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theory ReTest
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imports "Main"
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begin
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section {* Sequential Composition of Sets *}
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definition
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Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
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where
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"A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"
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fun spow where
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"spow s 0 = []"
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| "spow s (Suc n) = s @ spow s n"
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text {* Two Simple Properties about Sequential Composition *}
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lemma seq_empty [simp]:
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shows "A ;; {[]} = A"
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and "{[]} ;; A = A"
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by (simp_all add: Sequ_def)
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lemma seq_null [simp]:
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shows "A ;; {} = {}"
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and "{} ;; A = {}"
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by (simp_all add: Sequ_def)
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definition
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Der :: "char \<Rightarrow> string set \<Rightarrow> string set"
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where
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"Der c A \<equiv> {s. [c] @ s \<in> A}"
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definition
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Ders :: "string \<Rightarrow> string set \<Rightarrow> string set"
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where
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"Ders s A \<equiv> {s' | s'. s @ s' \<in> A}"
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lemma Der_null [simp]:
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shows "Der c {} = {}"
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unfolding Der_def
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by auto
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lemma Der_empty [simp]:
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shows "Der c {[]} = {}"
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unfolding Der_def
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by auto
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lemma Der_char [simp]:
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shows "Der c {[d]} = (if c = d then {[]} else {})"
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unfolding Der_def
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by auto
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lemma Der_union [simp]:
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shows "Der c (A \<union> B) = Der c A \<union> Der c B"
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unfolding Der_def
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by auto
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lemma Der_seq [simp]:
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shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"
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unfolding Der_def Sequ_def
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apply (auto simp add: Cons_eq_append_conv)
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done
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lemma seq_image:
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assumes "\<forall>s1 s2. f (s1 @ s2) = (f s1) @ (f s2)"
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shows "f ` (A ;; B) = (f ` A) ;; (f ` B)"
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apply(auto simp add: Sequ_def image_def)
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apply(rule_tac x="f s1" in exI)
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apply(rule_tac x="f s2" in exI)
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using assms
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apply(auto)
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apply(rule_tac x="xa @ xb" in exI)
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using assms
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apply(auto)
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done
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section {* Kleene Star for Sets *}
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inductive_set
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Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
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for A :: "string set"
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where
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start[intro]: "[] \<in> A\<star>"
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| step[intro]: "\<lbrakk>s1 \<in> A; s2 \<in> A\<star>\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> A\<star>"
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lemma star_cases:
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shows "A\<star> = {[]} \<union> A ;; A\<star>"
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unfolding Sequ_def
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by (auto) (metis Star.simps)
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fun
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pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [100,100] 100)
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where
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"A \<up> 0 = {[]}"
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| "A \<up> (Suc n) = A ;; (A \<up> n)"
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lemma star1:
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shows "s \<in> A\<star> \<Longrightarrow> \<exists>n. s \<in> A \<up> n"
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apply(induct rule: Star.induct)
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apply (metis pow.simps(1) insertI1)
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apply(auto)
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apply(rule_tac x="Suc n" in exI)
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apply(auto simp add: Sequ_def)
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done
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lemma star2:
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shows "s \<in> A \<up> n \<Longrightarrow> s \<in> A\<star>"
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apply(induct n arbitrary: s)
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apply (metis pow.simps(1) Star.simps empty_iff insertE)
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apply(auto simp add: Sequ_def)
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done
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lemma star3:
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shows "A\<star> = (\<Union>i. A \<up> i)"
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using star1 star2
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apply(auto)
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done
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lemma star4:
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shows "s \<in> A \<up> n \<Longrightarrow> \<exists>ss. s = concat ss \<and> (\<forall>s' \<in> set ss. s' \<in> A)"
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apply(induct n arbitrary: s)
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apply(auto simp add: Sequ_def)
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apply(rule_tac x="[]" in exI)
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apply(auto)
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apply(drule_tac x="s2" in meta_spec)
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apply(auto)
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by (metis concat.simps(2) insertE set_simps(2))
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lemma star5:
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assumes "f [] = []"
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assumes "\<forall>s1 s2. f (s1 @ s2) = (f s1) @ (f s2)"
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shows "(f ` A) \<up> n = f ` (A \<up> n)"
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apply(induct n)
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apply(simp add: assms)
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apply(simp)
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apply(subst seq_image[OF assms(2)])
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apply(simp)
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done
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lemma star6:
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assumes "f [] = []"
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assumes "\<forall>s1 s2. f (s1 @ s2) = (f s1) @ (f s2)"
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shows "(f ` A)\<star> = f ` (A\<star>)"
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apply(simp add: star3)
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apply(simp add: image_UN)
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apply(subst star5[OF assms])
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apply(simp)
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done
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lemma star_decomp:
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assumes a: "c # x \<in> A\<star>"
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shows "\<exists>a b. x = a @ b \<and> c # a \<in> A \<and> b \<in> A\<star>"
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using a
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by (induct x\<equiv>"c # x" rule: Star.induct)
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(auto simp add: append_eq_Cons_conv)
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lemma Der_star [simp]:
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shows "Der c (A\<star>) = (Der c A) ;; A\<star>"
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proof -
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have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"
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by (simp only: star_cases[symmetric])
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also have "... = Der c (A ;; A\<star>)"
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by (simp only: Der_union Der_empty) (simp)
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also have "... = (Der c A) ;; A\<star> \<union> (if [] \<in> A then Der c (A\<star>) else {})"
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by simp
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also have "... = (Der c A) ;; A\<star>"
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unfolding Sequ_def Der_def
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by (auto dest: star_decomp)
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finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .
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qed
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section {* Regular Expressions *}
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datatype rexp =
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NULL
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| EMPTY
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| CHAR char
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| SEQ rexp rexp
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| ALT rexp rexp
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| STAR rexp
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section {* Semantics of Regular Expressions *}
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fun
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L :: "rexp \<Rightarrow> string set"
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where
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"L (NULL) = {}"
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| "L (EMPTY) = {[]}"
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| "L (CHAR c) = {[c]}"
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| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
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| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
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| "L (STAR r) = (L r)\<star>"
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fun
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nullable :: "rexp \<Rightarrow> bool"
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where
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"nullable (NULL) = False"
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| "nullable (EMPTY) = True"
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| "nullable (CHAR c) = False"
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| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
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| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
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| "nullable (STAR r) = True"
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lemma nullable_correctness:
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shows "nullable r \<longleftrightarrow> [] \<in> (L r)"
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apply (induct r)
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apply(auto simp add: Sequ_def)
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done
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section {* Values *}
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datatype val =
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Void
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| Char char
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| Seq val val
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| Right val
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| Left val
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| Stars "val list"
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section {* The string behind a value *}
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fun
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flat :: "val \<Rightarrow> string"
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where
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"flat (Void) = []"
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| "flat (Char c) = [c]"
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| "flat (Left v) = flat v"
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| "flat (Right v) = flat v"
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| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
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| "flat (Stars []) = []"
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| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))"
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lemma [simp]:
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"flat (Stars vs) = concat (map flat vs)"
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apply(induct vs)
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apply(auto)
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done
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section {* Relation between values and regular expressions *}
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inductive
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NPrf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<Turnstile> _ : _" [100, 100] 100)
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where
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"\<lbrakk>\<Turnstile> v1 : r1; \<Turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<Turnstile> Seq v1 v2 : SEQ r1 r2"
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| "\<Turnstile> v1 : r1 \<Longrightarrow> \<Turnstile> Left v1 : ALT r1 r2"
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| "\<Turnstile> v2 : r2 \<Longrightarrow> \<Turnstile> Right v2 : ALT r1 r2"
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| "\<Turnstile> Void : EMPTY"
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| "\<Turnstile> Char c : CHAR c"
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| "\<Turnstile> Stars [] : STAR r"
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| "\<lbrakk>\<Turnstile> v : r; \<Turnstile> Stars vs : STAR r; flat v \<noteq> []\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (v # vs) : STAR r"
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inductive
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Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100)
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where
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"\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2"
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| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2"
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| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2"
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| "\<turnstile> Void : EMPTY"
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| "\<turnstile> Char c : CHAR c"
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| "\<turnstile> Stars [] : STAR r"
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| "\<lbrakk>\<turnstile> v : r; \<turnstile> Stars vs : STAR r\<rbrakk> \<Longrightarrow> \<turnstile> Stars (v # vs) : STAR r"
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lemma NPrf_imp_Prf:
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assumes "\<Turnstile> v : r"
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shows "\<turnstile> v : r"
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using assms
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apply(induct)
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apply(auto intro: Prf.intros)
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done
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lemma NPrf_Prf_val:
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shows "\<turnstile> v : r \<Longrightarrow> \<exists>v'. flat v' = flat v \<and> \<Turnstile> v' : r"
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and "\<turnstile> Stars vs : r \<Longrightarrow> \<exists>vs'. flat (Stars vs') = flat (Stars vs) \<and> \<Turnstile> Stars vs' : r"
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using assms
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apply(induct v and vs arbitrary: r and r rule: val.inducts)
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apply(auto)[1]
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apply(erule Prf.cases)
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apply(simp_all)[7]
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apply(rule_tac x="Void" in exI)
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apply(simp)
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apply(rule NPrf.intros)
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apply(erule Prf.cases)
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apply(simp_all)[7]
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apply(rule_tac x="Char c" in exI)
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apply(simp)
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apply(rule NPrf.intros)
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apply(erule Prf.cases)
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apply(simp_all)[7]
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apply(auto)[1]
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apply(drule_tac x="r1" in meta_spec)
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apply(drule_tac x="r2" in meta_spec)
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apply(simp)
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apply(auto)[1]
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apply(rule_tac x="Seq v' v'a" in exI)
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apply(simp)
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apply (metis NPrf.intros(1))
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apply(erule Prf.cases)
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apply(simp_all)[7]
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apply(clarify)
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apply(drule_tac x="r2" in meta_spec)
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apply(simp)
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apply(auto)[1]
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apply(rule_tac x="Right v'" in exI)
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apply(simp)
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apply (metis NPrf.intros)
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apply(erule Prf.cases)
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apply(simp_all)[7]
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apply(clarify)
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apply(drule_tac x="r1" in meta_spec)
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apply(simp)
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apply(auto)[1]
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apply(rule_tac x="Left v'" in exI)
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apply(simp)
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apply (metis NPrf.intros)
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apply(drule_tac x="r" in meta_spec)
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apply(simp)
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apply(auto)[1]
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apply(rule_tac x="Stars vs'" in exI)
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apply(simp)
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apply(rule_tac x="[]" in exI)
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apply(simp)
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apply(erule Prf.cases)
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apply(simp_all)[7]
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apply (metis NPrf.intros(6))
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apply(erule Prf.cases)
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|
334 |
apply(simp_all)[7]
|
|
|
335 |
apply(auto)[1]
|
|
|
336 |
apply(drule_tac x="ra" in meta_spec)
|
|
|
337 |
apply(simp)
|
|
|
338 |
apply(drule_tac x="STAR ra" in meta_spec)
|
|
|
339 |
apply(simp)
|
|
|
340 |
apply(auto)
|
|
|
341 |
apply(case_tac "flat v = []")
|
|
|
342 |
apply(rule_tac x="vs'" in exI)
|
|
|
343 |
apply(simp)
|
|
|
344 |
apply(rule_tac x="v' # vs'" in exI)
|
|
|
345 |
apply(simp)
|
|
|
346 |
apply(rule NPrf.intros)
|
|
|
347 |
apply(auto)
|
|
|
348 |
done
|
|
|
349 |
|
|
|
350 |
lemma NPrf_Prf:
|
|
|
351 |
shows "{flat v | v. \<turnstile> v : r} = {flat v | v. \<Turnstile> v : r}"
|
|
|
352 |
apply(auto)
|
|
|
353 |
apply (metis NPrf_Prf_val(1))
|
|
|
354 |
by (metis NPrf_imp_Prf)
|
|
|
355 |
|
|
|
356 |
|
|
|
357 |
lemma not_nullable_flat:
|
|
|
358 |
assumes "\<turnstile> v : r" "\<not>nullable r"
|
|
|
359 |
shows "flat v \<noteq> []"
|
|
|
360 |
using assms
|
|
|
361 |
apply(induct)
|
|
|
362 |
apply(auto)
|
|
|
363 |
done
|
|
|
364 |
|
|
|
365 |
lemma Prf_flat_L:
|
|
|
366 |
assumes "\<turnstile> v : r" shows "flat v \<in> L r"
|
|
|
367 |
using assms
|
|
|
368 |
apply(induct v r rule: Prf.induct)
|
|
|
369 |
apply(auto simp add: Sequ_def)
|
|
|
370 |
done
|
|
|
371 |
|
|
|
372 |
lemma NPrf_flat_L:
|
|
|
373 |
assumes "\<Turnstile> v : r" shows "flat v \<in> L r"
|
|
|
374 |
using assms
|
|
|
375 |
by (metis NPrf_imp_Prf Prf_flat_L)
|
|
|
376 |
|
|
|
377 |
lemma Prf_Stars:
|
|
|
378 |
assumes "\<forall>v \<in> set vs. \<turnstile> v : r"
|
|
|
379 |
shows "\<turnstile> Stars vs : STAR r"
|
|
|
380 |
using assms
|
|
|
381 |
apply(induct vs)
|
|
|
382 |
apply (metis Prf.intros(6))
|
|
|
383 |
by (metis Prf.intros(7) insert_iff set_simps(2))
|
|
|
384 |
|
|
|
385 |
lemma Star_string:
|
|
|
386 |
assumes "s \<in> A\<star>"
|
|
|
387 |
shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)"
|
|
|
388 |
using assms
|
|
|
389 |
apply(induct rule: Star.induct)
|
|
|
390 |
apply(auto)
|
|
|
391 |
apply(rule_tac x="[]" in exI)
|
|
|
392 |
apply(simp)
|
|
|
393 |
apply(rule_tac x="s1#ss" in exI)
|
|
|
394 |
apply(simp)
|
|
|
395 |
done
|
|
|
396 |
|
|
|
397 |
lemma Star_val:
|
|
|
398 |
assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
|
|
|
399 |
shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
|
|
|
400 |
using assms
|
|
|
401 |
apply(induct ss)
|
|
|
402 |
apply(auto)
|
|
|
403 |
apply (metis empty_iff list.set(1))
|
|
|
404 |
by (metis concat.simps(2) list.simps(9) set_ConsD)
|
|
|
405 |
|
|
|
406 |
lemma Star_valN:
|
|
|
407 |
assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
|
|
|
408 |
shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<Turnstile> v : r)"
|
|
|
409 |
using assms
|
|
|
410 |
apply(induct ss)
|
|
|
411 |
apply(auto)
|
|
|
412 |
apply (metis empty_iff list.set(1))
|
|
|
413 |
by (metis concat.simps(2) list.simps(9) set_ConsD)
|
|
|
414 |
|
|
|
415 |
lemma L_flat_Prf:
|
|
|
416 |
"L(r) = {flat v | v. \<turnstile> v : r}"
|
|
|
417 |
apply(induct r)
|
|
|
418 |
apply(auto dest: Prf_flat_L simp add: Sequ_def)
|
|
|
419 |
apply (metis Prf.intros(4) flat.simps(1))
|
|
|
420 |
apply (metis Prf.intros(5) flat.simps(2))
|
|
|
421 |
apply (metis Prf.intros(1) flat.simps(5))
|
|
|
422 |
apply (metis Prf.intros(2) flat.simps(3))
|
|
|
423 |
apply (metis Prf.intros(3) flat.simps(4))
|
|
|
424 |
apply(erule Prf.cases)
|
|
|
425 |
apply(auto)
|
|
|
426 |
apply(subgoal_tac "\<exists>vs::val list. concat (map flat vs) = x \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")
|
|
|
427 |
apply(auto)[1]
|
|
|
428 |
apply(rule_tac x="Stars vs" in exI)
|
|
|
429 |
apply(simp)
|
|
|
430 |
apply(rule Prf_Stars)
|
|
|
431 |
apply(simp)
|
|
|
432 |
apply(drule Star_string)
|
|
|
433 |
apply(auto)
|
|
|
434 |
apply(rule Star_val)
|
|
|
435 |
apply(simp)
|
|
|
436 |
done
|
|
|
437 |
|
|
|
438 |
lemma L_flat_NPrf:
|
|
|
439 |
"L(r) = {flat v | v. \<Turnstile> v : r}"
|
|
|
440 |
by (metis L_flat_Prf NPrf_Prf)
|
|
|
441 |
|
|
|
442 |
text {* nicer proofs by Fahad *}
|
|
|
443 |
|
|
|
444 |
lemma Prf_Star_flat_L:
|
|
|
445 |
assumes "\<turnstile> v : STAR r" shows "flat v \<in> (L r)\<star>"
|
|
|
446 |
using assms
|
|
|
447 |
apply(induct v r\<equiv>"STAR r" arbitrary: r rule: Prf.induct)
|
|
|
448 |
apply(auto)
|
|
|
449 |
apply(simp add: star3)
|
|
|
450 |
apply(auto)
|
|
|
451 |
apply(rule_tac x="Suc x" in exI)
|
|
|
452 |
apply(auto simp add: Sequ_def)
|
|
|
453 |
apply(rule_tac x="flat v" in exI)
|
|
|
454 |
apply(rule_tac x="flat (Stars vs)" in exI)
|
|
|
455 |
apply(auto)
|
|
|
456 |
by (metis Prf_flat_L)
|
|
|
457 |
|
|
|
458 |
lemma L_flat_Prf2:
|
|
|
459 |
"L(r) = {flat v | v. \<turnstile> v : r}"
|
|
|
460 |
apply(induct r)
|
|
|
461 |
apply(auto)
|
|
|
462 |
using L.simps(1) Prf_flat_L
|
|
|
463 |
apply(blast)
|
|
|
464 |
using Prf.intros(4)
|
|
|
465 |
apply(force)
|
|
|
466 |
using L.simps(2) Prf_flat_L
|
|
|
467 |
apply(blast)
|
|
|
468 |
using Prf.intros(5) apply force
|
|
|
469 |
using L.simps(3) Prf_flat_L apply blast
|
|
|
470 |
using L_flat_Prf apply auto[1]
|
|
|
471 |
apply (smt L.simps(4) Sequ_def mem_Collect_eq)
|
|
|
472 |
using Prf_flat_L
|
|
|
473 |
apply(fastforce)
|
|
|
474 |
apply(metis Prf.intros(2) flat.simps(3))
|
|
|
475 |
apply(metis Prf.intros(3) flat.simps(4))
|
|
|
476 |
apply(erule Prf.cases)
|
|
|
477 |
apply(simp)
|
|
|
478 |
apply(simp)
|
|
|
479 |
apply(auto)
|
|
|
480 |
using L_flat_Prf apply auto[1]
|
|
|
481 |
apply (smt Collect_cong L.simps(6) mem_Collect_eq)
|
|
|
482 |
using Prf_Star_flat_L
|
|
|
483 |
apply(fastforce)
|
|
|
484 |
done
|
|
|
485 |
|
|
|
486 |
|
|
|
487 |
section {* Values Sets *}
|
|
|
488 |
|
|
|
489 |
definition prefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubseteq> _" [100, 100] 100)
|
|
|
490 |
where
|
|
|
491 |
"s1 \<sqsubseteq> s2 \<equiv> \<exists>s3. s1 @ s3 = s2"
|
|
|
492 |
|
|
|
493 |
definition sprefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubset> _" [100, 100] 100)
|
|
|
494 |
where
|
|
|
495 |
"s1 \<sqsubset> s2 \<equiv> (s1 \<sqsubseteq> s2 \<and> s1 \<noteq> s2)"
|
|
|
496 |
|
|
|
497 |
lemma length_sprefix:
|
|
|
498 |
"s1 \<sqsubset> s2 \<Longrightarrow> length s1 < length s2"
|
|
|
499 |
unfolding sprefix_def prefix_def
|
|
|
500 |
by (auto)
|
|
|
501 |
|
|
|
502 |
definition Prefixes :: "string \<Rightarrow> string set" where
|
|
|
503 |
"Prefixes s \<equiv> {sp. sp \<sqsubseteq> s}"
|
|
|
504 |
|
|
|
505 |
definition Suffixes :: "string \<Rightarrow> string set" where
|
|
|
506 |
"Suffixes s \<equiv> rev ` (Prefixes (rev s))"
|
|
|
507 |
|
|
|
508 |
definition SPrefixes :: "string \<Rightarrow> string set" where
|
|
|
509 |
"SPrefixes s \<equiv> {sp. sp \<sqsubset> s}"
|
|
|
510 |
|
|
|
511 |
definition SSuffixes :: "string \<Rightarrow> string set" where
|
|
|
512 |
"SSuffixes s \<equiv> rev ` (SPrefixes (rev s))"
|
|
|
513 |
|
|
|
514 |
lemma Suffixes_in:
|
|
|
515 |
"\<exists>s1. s1 @ s2 = s3 \<Longrightarrow> s2 \<in> Suffixes s3"
|
|
|
516 |
unfolding Suffixes_def Prefixes_def prefix_def image_def
|
|
|
517 |
apply(auto)
|
|
|
518 |
by (metis rev_rev_ident)
|
|
|
519 |
|
|
|
520 |
lemma SSuffixes_in:
|
|
|
521 |
"\<exists>s1. s1 \<noteq> [] \<and> s1 @ s2 = s3 \<Longrightarrow> s2 \<in> SSuffixes s3"
|
|
|
522 |
unfolding SSuffixes_def Suffixes_def SPrefixes_def Prefixes_def sprefix_def prefix_def image_def
|
|
|
523 |
apply(auto)
|
|
|
524 |
by (metis append_self_conv rev.simps(1) rev_rev_ident)
|
|
|
525 |
|
|
|
526 |
lemma Prefixes_Cons:
|
|
|
527 |
"Prefixes (c # s) = {[]} \<union> {c # sp | sp. sp \<in> Prefixes s}"
|
|
|
528 |
unfolding Prefixes_def prefix_def
|
|
|
529 |
apply(auto simp add: append_eq_Cons_conv)
|
|
|
530 |
done
|
|
|
531 |
|
|
|
532 |
lemma finite_Prefixes:
|
|
|
533 |
"finite (Prefixes s)"
|
|
|
534 |
apply(induct s)
|
|
|
535 |
apply(auto simp add: Prefixes_def prefix_def)[1]
|
|
|
536 |
apply(simp add: Prefixes_Cons)
|
|
|
537 |
done
|
|
|
538 |
|
|
|
539 |
lemma finite_Suffixes:
|
|
|
540 |
"finite (Suffixes s)"
|
|
|
541 |
unfolding Suffixes_def
|
|
|
542 |
apply(rule finite_imageI)
|
|
|
543 |
apply(rule finite_Prefixes)
|
|
|
544 |
done
|
|
|
545 |
|
|
|
546 |
lemma prefix_Cons:
|
|
|
547 |
"((c # s1) \<sqsubseteq> (c # s2)) = (s1 \<sqsubseteq> s2)"
|
|
|
548 |
apply(auto simp add: prefix_def)
|
|
|
549 |
done
|
|
|
550 |
|
|
|
551 |
lemma prefix_append:
|
|
|
552 |
"((s @ s1) \<sqsubseteq> (s @ s2)) = (s1 \<sqsubseteq> s2)"
|
|
|
553 |
apply(induct s)
|
|
|
554 |
apply(simp)
|
|
|
555 |
apply(simp add: prefix_Cons)
|
|
|
556 |
done
|
|
|
557 |
|
|
|
558 |
|
|
|
559 |
definition Values :: "rexp \<Rightarrow> string \<Rightarrow> val set" where
|
|
|
560 |
"Values r s \<equiv> {v. \<turnstile> v : r \<and> flat v \<sqsubseteq> s}"
|
|
|
561 |
|
|
|
562 |
definition SValues :: "rexp \<Rightarrow> string \<Rightarrow> val set" where
|
|
|
563 |
"SValues r s \<equiv> {v. \<turnstile> v : r \<and> flat v = s}"
|
|
|
564 |
|
|
|
565 |
|
|
|
566 |
definition NValues :: "rexp \<Rightarrow> string \<Rightarrow> val set" where
|
|
|
567 |
"NValues r s \<equiv> {v. \<Turnstile> v : r \<and> flat v \<sqsubseteq> s}"
|
|
|
568 |
|
|
|
569 |
lemma NValues_STAR_Nil:
|
|
|
570 |
"NValues (STAR r) [] = {Stars []}"
|
|
|
571 |
apply(auto simp add: NValues_def prefix_def)
|
|
|
572 |
apply(erule NPrf.cases)
|
|
|
573 |
apply(auto)
|
|
|
574 |
by (metis NPrf.intros(6))
|
|
|
575 |
|
|
|
576 |
|
|
|
577 |
definition rest :: "val \<Rightarrow> string \<Rightarrow> string" where
|
|
|
578 |
"rest v s \<equiv> drop (length (flat v)) s"
|
|
|
579 |
|
|
|
580 |
lemma rest_Nil:
|
|
|
581 |
"rest v [] = []"
|
|
|
582 |
apply(simp add: rest_def)
|
|
|
583 |
done
|
|
|
584 |
|
|
|
585 |
lemma rest_Suffixes:
|
|
|
586 |
"rest v s \<in> Suffixes s"
|
|
|
587 |
unfolding rest_def
|
|
|
588 |
by (metis Suffixes_in append_take_drop_id)
|
|
|
589 |
|
|
|
590 |
lemma rest_SSuffixes:
|
|
|
591 |
assumes "flat v \<noteq> []" "s \<noteq> []"
|
|
|
592 |
shows "rest v s \<in> SSuffixes s"
|
|
|
593 |
using assms
|
|
|
594 |
unfolding rest_def
|
|
|
595 |
thm SSuffixes_in
|
|
|
596 |
apply(rule_tac SSuffixes_in)
|
|
|
597 |
apply(rule_tac x="take (length (flat v)) s" in exI)
|
|
|
598 |
apply(simp add: sprefix_def)
|
|
|
599 |
done
|
|
|
600 |
|
|
|
601 |
|
|
|
602 |
lemma Values_recs:
|
|
|
603 |
"Values (NULL) s = {}"
|
|
|
604 |
"Values (EMPTY) s = {Void}"
|
|
|
605 |
"Values (CHAR c) s = (if [c] \<sqsubseteq> s then {Char c} else {})"
|
|
|
606 |
"Values (ALT r1 r2) s = {Left v | v. v \<in> Values r1 s} \<union> {Right v | v. v \<in> Values r2 s}"
|
|
|
607 |
"Values (SEQ r1 r2) s = {Seq v1 v2 | v1 v2. v1 \<in> Values r1 s \<and> v2 \<in> Values r2 (rest v1 s)}"
|
|
|
608 |
"Values (STAR r) s =
|
|
|
609 |
{Stars []} \<union> {Stars (v # vs) | v vs. v \<in> Values r s \<and> Stars vs \<in> Values (STAR r) (rest v s)}"
|
|
|
610 |
unfolding Values_def
|
|
|
611 |
apply(auto)
|
|
|
612 |
(*NULL*)
|
|
|
613 |
apply(erule Prf.cases)
|
|
|
614 |
apply(simp_all)[7]
|
|
|
615 |
(*EMPTY*)
|
|
|
616 |
apply(erule Prf.cases)
|
|
|
617 |
apply(simp_all)[7]
|
|
|
618 |
apply(rule Prf.intros)
|
|
|
619 |
apply (metis append_Nil prefix_def)
|
|
|
620 |
(*CHAR*)
|
|
|
621 |
apply(erule Prf.cases)
|
|
|
622 |
apply(simp_all)[7]
|
|
|
623 |
apply(rule Prf.intros)
|
|
|
624 |
apply(erule Prf.cases)
|
|
|
625 |
apply(simp_all)[7]
|
|
|
626 |
(*ALT*)
|
|
|
627 |
apply(erule Prf.cases)
|
|
|
628 |
apply(simp_all)[7]
|
|
|
629 |
apply (metis Prf.intros(2))
|
|
|
630 |
apply (metis Prf.intros(3))
|
|
|
631 |
(*SEQ*)
|
|
|
632 |
apply(erule Prf.cases)
|
|
|
633 |
apply(simp_all)[7]
|
|
|
634 |
apply (simp add: append_eq_conv_conj prefix_def rest_def)
|
|
|
635 |
apply (metis Prf.intros(1))
|
|
|
636 |
apply (simp add: append_eq_conv_conj prefix_def rest_def)
|
|
|
637 |
(*STAR*)
|
|
|
638 |
apply(erule Prf.cases)
|
|
|
639 |
apply(simp_all)[7]
|
|
|
640 |
apply(rule conjI)
|
|
|
641 |
apply(simp add: prefix_def)
|
|
|
642 |
apply(auto)[1]
|
|
|
643 |
apply(simp add: prefix_def)
|
|
|
644 |
apply(auto)[1]
|
|
|
645 |
apply (metis append_eq_conv_conj rest_def)
|
|
|
646 |
apply (metis Prf.intros(6))
|
|
|
647 |
apply (metis append_Nil prefix_def)
|
|
|
648 |
apply (metis Prf.intros(7))
|
|
|
649 |
by (metis append_eq_conv_conj prefix_append prefix_def rest_def)
|
|
|
650 |
|
|
|
651 |
lemma NValues_recs:
|
|
|
652 |
"NValues (NULL) s = {}"
|
|
|
653 |
"NValues (EMPTY) s = {Void}"
|
|
|
654 |
"NValues (CHAR c) s = (if [c] \<sqsubseteq> s then {Char c} else {})"
|
|
|
655 |
"NValues (ALT r1 r2) s = {Left v | v. v \<in> NValues r1 s} \<union> {Right v | v. v \<in> NValues r2 s}"
|
|
|
656 |
"NValues (SEQ r1 r2) s = {Seq v1 v2 | v1 v2. v1 \<in> NValues r1 s \<and> v2 \<in> NValues r2 (rest v1 s)}"
|
|
|
657 |
"NValues (STAR r) s =
|
|
|
658 |
{Stars []} \<union> {Stars (v # vs) | v vs. v \<in> NValues r s \<and> flat v \<noteq> [] \<and> Stars vs \<in> NValues (STAR r) (rest v s)}"
|
|
|
659 |
unfolding NValues_def
|
|
|
660 |
apply(auto)
|
|
|
661 |
(*NULL*)
|
|
|
662 |
apply(erule NPrf.cases)
|
|
|
663 |
apply(simp_all)[7]
|
|
|
664 |
(*EMPTY*)
|
|
|
665 |
apply(erule NPrf.cases)
|
|
|
666 |
apply(simp_all)[7]
|
|
|
667 |
apply(rule NPrf.intros)
|
|
|
668 |
apply (metis append_Nil prefix_def)
|
|
|
669 |
(*CHAR*)
|
|
|
670 |
apply(erule NPrf.cases)
|
|
|
671 |
apply(simp_all)[7]
|
|
|
672 |
apply(rule NPrf.intros)
|
|
|
673 |
apply(erule NPrf.cases)
|
|
|
674 |
apply(simp_all)[7]
|
|
|
675 |
(*ALT*)
|
|
|
676 |
apply(erule NPrf.cases)
|
|
|
677 |
apply(simp_all)[7]
|
|
|
678 |
apply (metis NPrf.intros(2))
|
|
|
679 |
apply (metis NPrf.intros(3))
|
|
|
680 |
(*SEQ*)
|
|
|
681 |
apply(erule NPrf.cases)
|
|
|
682 |
apply(simp_all)[7]
|
|
|
683 |
apply (simp add: append_eq_conv_conj prefix_def rest_def)
|
|
|
684 |
apply (metis NPrf.intros(1))
|
|
|
685 |
apply (simp add: append_eq_conv_conj prefix_def rest_def)
|
|
|
686 |
(*STAR*)
|
|
|
687 |
apply(erule NPrf.cases)
|
|
|
688 |
apply(simp_all)
|
|
|
689 |
apply(rule conjI)
|
|
|
690 |
apply(simp add: prefix_def)
|
|
|
691 |
apply(auto)[1]
|
|
|
692 |
apply(simp add: prefix_def)
|
|
|
693 |
apply(auto)[1]
|
|
|
694 |
apply (metis append_eq_conv_conj rest_def)
|
|
|
695 |
apply (metis NPrf.intros(6))
|
|
|
696 |
apply (metis append_Nil prefix_def)
|
|
|
697 |
apply (metis NPrf.intros(7))
|
|
|
698 |
by (metis append_eq_conv_conj prefix_append prefix_def rest_def)
|
|
|
699 |
|
|
|
700 |
lemma SValues_recs:
|
|
|
701 |
"SValues (NULL) s = {}"
|
|
|
702 |
"SValues (EMPTY) s = (if s = [] then {Void} else {})"
|
|
|
703 |
"SValues (CHAR c) s = (if [c] = s then {Char c} else {})"
|
|
|
704 |
"SValues (ALT r1 r2) s = {Left v | v. v \<in> SValues r1 s} \<union> {Right v | v. v \<in> SValues r2 s}"
|
|
|
705 |
"SValues (SEQ r1 r2) s = {Seq v1 v2 | v1 v2. \<exists>s1 s2. s = s1 @ s2 \<and> v1 \<in> SValues r1 s1 \<and> v2 \<in> SValues r2 s2}"
|
|
|
706 |
"SValues (STAR r) s = (if s = [] then {Stars []} else {}) \<union>
|
|
|
707 |
{Stars (v # vs) | v vs. \<exists>s1 s2. s = s1 @ s2 \<and> v \<in> SValues r s1 \<and> Stars vs \<in> SValues (STAR r) s2}"
|
|
|
708 |
unfolding SValues_def
|
|
|
709 |
apply(auto)
|
|
|
710 |
(*NULL*)
|
|
|
711 |
apply(erule Prf.cases)
|
|
|
712 |
apply(simp_all)[7]
|
|
|
713 |
(*EMPTY*)
|
|
|
714 |
apply(erule Prf.cases)
|
|
|
715 |
apply(simp_all)[7]
|
|
|
716 |
apply(rule Prf.intros)
|
|
|
717 |
apply(erule Prf.cases)
|
|
|
718 |
apply(simp_all)[7]
|
|
|
719 |
(*CHAR*)
|
|
|
720 |
apply(erule Prf.cases)
|
|
|
721 |
apply(simp_all)[7]
|
|
|
722 |
apply (metis Prf.intros(5))
|
|
|
723 |
apply(erule Prf.cases)
|
|
|
724 |
apply(simp_all)[7]
|
|
|
725 |
(*ALT*)
|
|
|
726 |
apply(erule Prf.cases)
|
|
|
727 |
apply(simp_all)[7]
|
|
|
728 |
apply metis
|
|
|
729 |
apply(erule Prf.intros)
|
|
|
730 |
apply(erule Prf.intros)
|
|
|
731 |
(* SEQ case *)
|
|
|
732 |
apply(erule Prf.cases)
|
|
|
733 |
apply(simp_all)[7]
|
|
|
734 |
apply (metis Prf.intros(1))
|
|
|
735 |
(* STAR case *)
|
|
|
736 |
apply(erule Prf.cases)
|
|
|
737 |
apply(simp_all)[7]
|
|
|
738 |
apply(rule Prf.intros)
|
|
|
739 |
apply (metis Prf.intros(7))
|
|
|
740 |
apply(erule Prf.cases)
|
|
|
741 |
apply(simp_all)[7]
|
|
|
742 |
apply (metis Prf.intros(7))
|
|
|
743 |
by (metis Prf.intros(7))
|
|
|
744 |
|
|
|
745 |
lemma finite_image_set2:
|
|
|
746 |
"finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {(x, y) | x y. P x \<and> Q y}"
|
|
|
747 |
by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {(x, y)}"]) auto
|
|
|
748 |
|
|
|
749 |
|
|
|
750 |
lemma NValues_finite_aux:
|
|
|
751 |
"(\<lambda>(r, s). finite (NValues r s)) (r, s)"
|
|
|
752 |
apply(rule wf_induct[of "measure size <*lex*> measure length",where P="\<lambda>(r, s). finite (NValues r s)"])
|
|
|
753 |
apply (metis wf_lex_prod wf_measure)
|
|
|
754 |
apply(auto)
|
|
|
755 |
apply(case_tac a)
|
|
|
756 |
apply(simp_all)
|
|
|
757 |
apply(simp add: NValues_recs)
|
|
|
758 |
apply(simp add: NValues_recs)
|
|
|
759 |
apply(simp add: NValues_recs)
|
|
|
760 |
apply(simp add: NValues_recs)
|
|
|
761 |
apply(rule_tac f="\<lambda>(x, y). Seq x y" and
|
|
|
762 |
A="{(v1, v2) | v1 v2. v1 \<in> NValues rexp1 b \<and> v2 \<in> NValues rexp2 (rest v1 b)}" in finite_surj)
|
|
|
763 |
prefer 2
|
|
|
764 |
apply(auto)[1]
|
|
|
765 |
apply(rule_tac B="\<Union>sp \<in> Suffixes b. {(v1, v2). v1 \<in> NValues rexp1 b \<and> v2 \<in> NValues rexp2 sp}" in finite_subset)
|
|
|
766 |
apply(auto)[1]
|
|
|
767 |
apply (metis rest_Suffixes)
|
|
|
768 |
apply(rule finite_UN_I)
|
|
|
769 |
apply(rule finite_Suffixes)
|
|
|
770 |
apply(simp)
|
|
|
771 |
apply(simp add: NValues_recs)
|
|
|
772 |
apply(clarify)
|
|
|
773 |
apply(subst NValues_recs)
|
|
|
774 |
apply(simp)
|
|
|
775 |
apply(rule_tac f="\<lambda>(v, vs). Stars (v # vs)" and
|
|
|
776 |
A="{(v, vs) | v vs. v \<in> NValues rexp b \<and> (flat v \<noteq> [] \<and> Stars vs \<in> NValues (STAR rexp) (rest v b))}" in finite_surj)
|
|
|
777 |
prefer 2
|
|
|
778 |
apply(auto)[1]
|
|
|
779 |
apply(auto)
|
|
|
780 |
apply(case_tac b)
|
|
|
781 |
apply(simp)
|
|
|
782 |
defer
|
|
|
783 |
apply(rule_tac B="\<Union>sp \<in> SSuffixes b. {(v, vs) | v vs. v \<in> NValues rexp b \<and> Stars vs \<in> NValues (STAR rexp) sp}" in finite_subset)
|
|
|
784 |
apply(auto)[1]
|
|
|
785 |
apply(rule_tac x="rest aa (a # list)" in bexI)
|
|
|
786 |
apply(simp)
|
|
|
787 |
apply (rule rest_SSuffixes)
|
|
|
788 |
apply(simp)
|
|
|
789 |
apply(simp)
|
|
|
790 |
apply(rule finite_UN_I)
|
|
|
791 |
defer
|
|
|
792 |
apply(frule_tac x="rexp" in spec)
|
|
|
793 |
apply(drule_tac x="b" in spec)
|
|
|
794 |
apply(drule conjunct1)
|
|
|
795 |
apply(drule mp)
|
|
|
796 |
apply(simp)
|
|
|
797 |
apply(drule_tac x="STAR rexp" in spec)
|
|
|
798 |
apply(drule_tac x="sp" in spec)
|
|
|
799 |
apply(drule conjunct2)
|
|
|
800 |
apply(drule mp)
|
|
|
801 |
apply(simp)
|
|
|
802 |
apply(simp add: prefix_def SPrefixes_def SSuffixes_def)
|
|
|
803 |
apply(auto)[1]
|
|
|
804 |
apply (metis length_Cons length_rev length_sprefix rev.simps(2))
|
|
|
805 |
apply(simp)
|
|
|
806 |
apply(rule finite_cartesian_product)
|
|
|
807 |
apply(simp)
|
|
|
808 |
apply(rule_tac f="Stars" in finite_imageD)
|
|
|
809 |
prefer 2
|
|
|
810 |
apply(auto simp add: inj_on_def)[1]
|
|
|
811 |
apply (metis finite_subset image_Collect_subsetI)
|
|
|
812 |
apply(simp add: rest_Nil)
|
|
|
813 |
apply(simp add: NValues_STAR_Nil)
|
|
|
814 |
apply(rule_tac B="{(v, vs). v \<in> NValues rexp [] \<and> vs = []}" in finite_subset)
|
|
|
815 |
apply(auto)[1]
|
|
|
816 |
apply(simp)
|
|
|
817 |
apply(rule_tac B="Suffixes b" in finite_subset)
|
|
|
818 |
apply(auto simp add: SSuffixes_def Suffixes_def Prefixes_def SPrefixes_def sprefix_def)[1]
|
|
|
819 |
by (metis finite_Suffixes)
|
|
|
820 |
|
|
|
821 |
lemma NValues_finite:
|
|
|
822 |
"finite (NValues r s)"
|
|
|
823 |
using NValues_finite_aux
|
|
|
824 |
apply(simp)
|
|
|
825 |
done
|
|
|
826 |
|
|
|
827 |
section {* Sulzmann functions *}
|
|
|
828 |
|
|
|
829 |
fun
|
|
|
830 |
mkeps :: "rexp \<Rightarrow> val"
|
|
|
831 |
where
|
|
|
832 |
"mkeps(EMPTY) = Void"
|
|
|
833 |
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
|
|
|
834 |
| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
|
|
|
835 |
| "mkeps(STAR r) = Stars []"
|
|
|
836 |
|
|
|
837 |
section {* Derivatives *}
|
|
|
838 |
|
|
|
839 |
fun
|
|
|
840 |
der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
|
|
|
841 |
where
|
|
|
842 |
"der c (NULL) = NULL"
|
|
|
843 |
| "der c (EMPTY) = NULL"
|
|
|
844 |
| "der c (CHAR c') = (if c = c' then EMPTY else NULL)"
|
|
|
845 |
| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"
|
|
|
846 |
| "der c (SEQ r1 r2) =
|
|
|
847 |
(if nullable r1
|
|
|
848 |
then ALT (SEQ (der c r1) r2) (der c r2)
|
|
|
849 |
else SEQ (der c r1) r2)"
|
|
|
850 |
| "der c (STAR r) = SEQ (der c r) (STAR r)"
|
|
|
851 |
|
|
|
852 |
fun
|
|
|
853 |
ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
|
|
|
854 |
where
|
|
|
855 |
"ders [] r = r"
|
|
|
856 |
| "ders (c # s) r = ders s (der c r)"
|
|
|
857 |
|
|
|
858 |
|
|
|
859 |
lemma der_correctness:
|
|
|
860 |
shows "L (der c r) = Der c (L r)"
|
|
|
861 |
apply(induct r)
|
|
|
862 |
apply(simp_all add: nullable_correctness)
|
|
|
863 |
done
|
|
|
864 |
|
|
|
865 |
lemma ders_correctness:
|
|
|
866 |
shows "L (ders s r) = Ders s (L r)"
|
|
|
867 |
apply(induct s arbitrary: r)
|
|
|
868 |
apply(simp add: Ders_def)
|
|
|
869 |
apply(simp)
|
|
|
870 |
apply(subst der_correctness)
|
|
|
871 |
apply(simp add: Ders_def Der_def)
|
|
|
872 |
done
|
|
|
873 |
|
|
|
874 |
section {* Injection function *}
|
|
|
875 |
|
|
|
876 |
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
|
|
|
877 |
where
|
|
|
878 |
"injval (CHAR d) c Void = Char d"
|
|
|
879 |
| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)"
|
|
|
880 |
| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)"
|
|
|
881 |
| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
|
|
|
882 |
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
|
|
|
883 |
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
|
|
|
884 |
| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
|
|
|
885 |
|
|
|
886 |
fun
|
|
|
887 |
lex :: "rexp \<Rightarrow> string \<Rightarrow> val option"
|
|
|
888 |
where
|
|
|
889 |
"lex r [] = (if nullable r then Some(mkeps r) else None)"
|
|
|
890 |
| "lex r (c#s) = (case (lex (der c r) s) of
|
|
|
891 |
None \<Rightarrow> None
|
|
|
892 |
| Some(v) \<Rightarrow> Some(injval r c v))"
|
|
|
893 |
|
|
|
894 |
fun
|
|
|
895 |
lex2 :: "rexp \<Rightarrow> string \<Rightarrow> val"
|
|
|
896 |
where
|
|
|
897 |
"lex2 r [] = mkeps r"
|
|
|
898 |
| "lex2 r (c#s) = injval r c (lex2 (der c r) s)"
|
|
|
899 |
|
|
|
900 |
|
|
|
901 |
section {* Projection function *}
|
|
|
902 |
|
|
|
903 |
fun projval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
|
|
|
904 |
where
|
|
|
905 |
"projval (CHAR d) c _ = Void"
|
|
|
906 |
| "projval (ALT r1 r2) c (Left v1) = Left (projval r1 c v1)"
|
|
|
907 |
| "projval (ALT r1 r2) c (Right v2) = Right (projval r2 c v2)"
|
|
|
908 |
| "projval (SEQ r1 r2) c (Seq v1 v2) =
|
|
|
909 |
(if flat v1 = [] then Right(projval r2 c v2)
|
|
|
910 |
else if nullable r1 then Left (Seq (projval r1 c v1) v2)
|
|
|
911 |
else Seq (projval r1 c v1) v2)"
|
|
|
912 |
| "projval (STAR r) c (Stars (v # vs)) = Seq (projval r c v) (Stars vs)"
|
|
|
913 |
|
|
|
914 |
|
|
|
915 |
|
|
|
916 |
lemma mkeps_nullable:
|
|
|
917 |
assumes "nullable(r)"
|
|
|
918 |
shows "\<turnstile> mkeps r : r"
|
|
|
919 |
using assms
|
|
|
920 |
apply(induct rule: nullable.induct)
|
|
|
921 |
apply(auto intro: Prf.intros)
|
|
|
922 |
done
|
|
|
923 |
|
|
|
924 |
lemma mkeps_flat:
|
|
|
925 |
assumes "nullable(r)"
|
|
|
926 |
shows "flat (mkeps r) = []"
|
|
|
927 |
using assms
|
|
|
928 |
apply(induct rule: nullable.induct)
|
|
|
929 |
apply(auto)
|
|
|
930 |
done
|
|
|
931 |
|
|
|
932 |
|
|
|
933 |
lemma v3:
|
|
|
934 |
assumes "\<turnstile> v : der c r"
|
|
|
935 |
shows "\<turnstile> (injval r c v) : r"
|
|
|
936 |
using assms
|
|
|
937 |
apply(induct arbitrary: v rule: der.induct)
|
|
|
938 |
apply(simp)
|
|
|
939 |
apply(erule Prf.cases)
|
|
|
940 |
apply(simp_all)[7]
|
|
|
941 |
apply(simp)
|
|
|
942 |
apply(erule Prf.cases)
|
|
|
943 |
apply(simp_all)[7]
|
|
|
944 |
apply(case_tac "c = c'")
|
|
|
945 |
apply(simp)
|
|
|
946 |
apply(erule Prf.cases)
|
|
|
947 |
apply(simp_all)[7]
|
|
|
948 |
apply (metis Prf.intros(5))
|
|
|
949 |
apply(simp)
|
|
|
950 |
apply(erule Prf.cases)
|
|
|
951 |
apply(simp_all)[7]
|
|
|
952 |
apply(simp)
|
|
|
953 |
apply(erule Prf.cases)
|
|
|
954 |
apply(simp_all)[7]
|
|
|
955 |
apply (metis Prf.intros(2))
|
|
|
956 |
apply (metis Prf.intros(3))
|
|
|
957 |
apply(simp)
|
|
|
958 |
apply(case_tac "nullable r1")
|
|
|
959 |
apply(simp)
|
|
|
960 |
apply(erule Prf.cases)
|
|
|
961 |
apply(simp_all)[7]
|
|
|
962 |
apply(auto)[1]
|
|
|
963 |
apply(erule Prf.cases)
|
|
|
964 |
apply(simp_all)[7]
|
|
|
965 |
apply(auto)[1]
|
|
|
966 |
apply (metis Prf.intros(1))
|
|
|
967 |
apply(auto)[1]
|
|
|
968 |
apply (metis Prf.intros(1) mkeps_nullable)
|
|
|
969 |
apply(simp)
|
|
|
970 |
apply(erule Prf.cases)
|
|
|
971 |
apply(simp_all)[7]
|
|
|
972 |
apply(auto)[1]
|
|
|
973 |
apply(rule Prf.intros)
|
|
|
974 |
apply(auto)[2]
|
|
|
975 |
apply(simp)
|
|
|
976 |
apply(erule Prf.cases)
|
|
|
977 |
apply(simp_all)[7]
|
|
|
978 |
apply(clarify)
|
|
|
979 |
apply(rotate_tac 2)
|
|
|
980 |
apply(erule Prf.cases)
|
|
|
981 |
apply(simp_all)[7]
|
|
|
982 |
apply(auto)
|
|
|
983 |
apply (metis Prf.intros(6) Prf.intros(7))
|
|
|
984 |
by (metis Prf.intros(7))
|
|
|
985 |
|
|
|
986 |
lemma v3_proj:
|
|
|
987 |
assumes "\<Turnstile> v : r" and "\<exists>s. (flat v) = c # s"
|
|
|
988 |
shows "\<Turnstile> (projval r c v) : der c r"
|
|
|
989 |
using assms
|
|
|
990 |
apply(induct rule: NPrf.induct)
|
|
|
991 |
prefer 4
|
|
|
992 |
apply(simp)
|
|
|
993 |
prefer 4
|
|
|
994 |
apply(simp)
|
|
|
995 |
apply (metis NPrf.intros(4))
|
|
|
996 |
prefer 2
|
|
|
997 |
apply(simp)
|
|
|
998 |
apply (metis NPrf.intros(2))
|
|
|
999 |
prefer 2
|
|
|
1000 |
apply(simp)
|
|
|
1001 |
apply (metis NPrf.intros(3))
|
|
|
1002 |
apply(auto)
|
|
|
1003 |
apply(rule NPrf.intros)
|
|
|
1004 |
apply(simp)
|
|
|
1005 |
apply (metis NPrf_imp_Prf not_nullable_flat)
|
|
|
1006 |
apply(rule NPrf.intros)
|
|
|
1007 |
apply(rule NPrf.intros)
|
|
|
1008 |
apply (metis Cons_eq_append_conv)
|
|
|
1009 |
apply(simp)
|
|
|
1010 |
apply(rule NPrf.intros)
|
|
|
1011 |
apply (metis Cons_eq_append_conv)
|
|
|
1012 |
apply(simp)
|
|
|
1013 |
(* Stars case *)
|
|
|
1014 |
apply(rule NPrf.intros)
|
|
|
1015 |
apply (metis Cons_eq_append_conv)
|
|
|
1016 |
apply(auto)
|
|
|
1017 |
done
|
|
|
1018 |
|
|
|
1019 |
lemma v4:
|
|
|
1020 |
assumes "\<turnstile> v : der c r"
|
|
|
1021 |
shows "flat (injval r c v) = c # (flat v)"
|
|
|
1022 |
using assms
|
|
|
1023 |
apply(induct arbitrary: v rule: der.induct)
|
|
|
1024 |
apply(simp)
|
|
|
1025 |
apply(erule Prf.cases)
|
|
|
1026 |
apply(simp_all)[7]
|
|
|
1027 |
apply(simp)
|
|
|
1028 |
apply(erule Prf.cases)
|
|
|
1029 |
apply(simp_all)[7]
|
|
|
1030 |
apply(simp)
|
|
|
1031 |
apply(case_tac "c = c'")
|
|
|
1032 |
apply(simp)
|
|
|
1033 |
apply(auto)[1]
|
|
|
1034 |
apply(erule Prf.cases)
|
|
|
1035 |
apply(simp_all)[7]
|
|
|
1036 |
apply(simp)
|
|
|
1037 |
apply(erule Prf.cases)
|
|
|
1038 |
apply(simp_all)[7]
|
|
|
1039 |
apply(simp)
|
|
|
1040 |
apply(erule Prf.cases)
|
|
|
1041 |
apply(simp_all)[7]
|
|
|
1042 |
apply(simp)
|
|
|
1043 |
apply(case_tac "nullable r1")
|
|
|
1044 |
apply(simp)
|
|
|
1045 |
apply(erule Prf.cases)
|
|
|
1046 |
apply(simp_all (no_asm_use))[7]
|
|
|
1047 |
apply(auto)[1]
|
|
|
1048 |
apply(erule Prf.cases)
|
|
|
1049 |
apply(simp_all)[7]
|
|
|
1050 |
apply(clarify)
|
|
|
1051 |
apply(simp only: injval.simps flat.simps)
|
|
|
1052 |
apply(auto)[1]
|
|
|
1053 |
apply (metis mkeps_flat)
|
|
|
1054 |
apply(simp)
|
|
|
1055 |
apply(erule Prf.cases)
|
|
|
1056 |
apply(simp_all)[7]
|
|
|
1057 |
apply(simp)
|
|
|
1058 |
apply(erule Prf.cases)
|
|
|
1059 |
apply(simp_all)[7]
|
|
|
1060 |
apply(auto)
|
|
|
1061 |
apply(rotate_tac 2)
|
|
|
1062 |
apply(erule Prf.cases)
|
|
|
1063 |
apply(simp_all)[7]
|
|
|
1064 |
done
|
|
|
1065 |
|
|
|
1066 |
lemma v4_proj:
|
|
|
1067 |
assumes "\<Turnstile> v : r" and "\<exists>s. (flat v) = c # s"
|
|
|
1068 |
shows "c # flat (projval r c v) = flat v"
|
|
|
1069 |
using assms
|
|
|
1070 |
apply(induct rule: NPrf.induct)
|
|
|
1071 |
prefer 4
|
|
|
1072 |
apply(simp)
|
|
|
1073 |
prefer 4
|
|
|
1074 |
apply(simp)
|
|
|
1075 |
prefer 2
|
|
|
1076 |
apply(simp)
|
|
|
1077 |
prefer 2
|
|
|
1078 |
apply(simp)
|
|
|
1079 |
apply(auto)
|
|
|
1080 |
apply (metis Cons_eq_append_conv)
|
|
|
1081 |
apply(simp add: append_eq_Cons_conv)
|
|
|
1082 |
apply(auto)
|
|
|
1083 |
done
|
|
|
1084 |
|
|
|
1085 |
lemma v4_proj2:
|
|
|
1086 |
assumes "\<Turnstile> v : r" and "(flat v) = c # s"
|
|
|
1087 |
shows "flat (projval r c v) = s"
|
|
|
1088 |
using assms
|
|
|
1089 |
by (metis list.inject v4_proj)
|
|
|
1090 |
|
|
|
1091 |
|
|
|
1092 |
definition
|
|
|
1093 |
PC31 :: "string \<Rightarrow> rexp \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
1094 |
where
|
|
|
1095 |
"PC31 s r r' \<equiv> s \<notin> L r"
|
|
|
1096 |
|
|
|
1097 |
definition
|
|
|
1098 |
PC41 :: "string \<Rightarrow> string \<Rightarrow> rexp \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
1099 |
where
|
|
|
1100 |
"PC41 s s' r r' \<equiv> (\<forall>x. (s @ x \<in> L r \<longrightarrow> s' \<in> {x} ;; L r' \<longrightarrow> x = []))"
|
|
|
1101 |
|
|
|
1102 |
|
|
|
1103 |
lemma
|
|
|
1104 |
L1: "\<not>(nullable r1) \<longrightarrow> [] \<in> L r2 \<longrightarrow> PC31 [] r1 r2" and
|
|
|
1105 |
L2: "s1 \<in> L(r1) \<longrightarrow> [] \<in> L(r2) \<longrightarrow> PC41 s1 [] r1 r2" and
|
|
|
1106 |
L3: "s2 \<in> L(der c r2) \<longrightarrow> PC31 s2 (der c r1) (der c r2) \<longrightarrow> PC31 (c#s2) r1 r2" and
|
|
|
1107 |
L4: "s1 \<in> L(der c r1) \<longrightarrow> s2 \<in> L(r2) \<longrightarrow> PC41 s1 s2 (der c r1) r2 \<longrightarrow> PC41 (c#s1) s2 r1 r2" and
|
|
|
1108 |
L5: "nullable(r1) \<longrightarrow> s2 \<in> L(der c r2) \<longrightarrow> PC31 s2 (SEQ (der c r1) r2) (der c r2) \<longrightarrow> PC41 [] (c#s2) r1 r2" and
|
|
|
1109 |
L6: "s0 \<in> L(der c r0) \<longrightarrow> s \<in> L(STAR r0) \<longrightarrow> PC41 s0 s (der c r0) (STAR r0) \<longrightarrow> PC41 (c#s0) s r0 (STAR r0)" and
|
|
|
1110 |
L7: "s' \<in> L(r') \<longrightarrow> s' \<in> L(r) \<longrightarrow> \<not>PC31 s' r r'" and
|
|
|
1111 |
L8: "s \<in> L(r) \<longrightarrow> s' \<in> L(r') \<longrightarrow> s @ x \<in> L(r) \<longrightarrow> s' \<in> {x} ;; (L(r') ;; {y}) \<longrightarrow> x \<noteq> [] \<longrightarrow> \<not>PC41 s s' r r'"
|
|
|
1112 |
apply(auto simp add: PC31_def PC41_def)[1]
|
|
|
1113 |
apply (metis nullable_correctness)
|
|
|
1114 |
apply(auto simp add: PC31_def PC41_def)[1]
|
|
|
1115 |
apply(simp add: Sequ_def)
|
|
|
1116 |
apply(auto simp add: PC31_def PC41_def)[1]
|
|
|
1117 |
apply(simp add: der_correctness Der_def)
|
|
|
1118 |
apply(auto simp add: PC31_def PC41_def)[1]
|
|
|
1119 |
apply(simp add: der_correctness Der_def Sequ_def)
|
|
|
1120 |
apply(auto simp add: PC31_def PC41_def)[1]
|
|
|
1121 |
apply(simp add: Sequ_def)
|
|
|
1122 |
apply(simp add: der_correctness Der_def)
|
|
|
1123 |
apply(auto)[1]
|
|
|
1124 |
apply (metis append_eq_Cons_conv)
|
|
|
1125 |
apply(auto simp add: PC31_def PC41_def)[1]
|
|
|
1126 |
apply(simp add: Sequ_def)
|
|
|
1127 |
apply(simp add: der_correctness Der_def)
|
|
|
1128 |
apply(auto simp add: PC31_def PC41_def)[1]
|
|
|
1129 |
apply(rule impI)+
|
|
|
1130 |
apply(rule notI)
|
|
|
1131 |
(* 8 fails *)
|
|
|
1132 |
oops
|
|
|
1133 |
|
|
|
1134 |
definition
|
|
|
1135 |
PC32 :: "string \<Rightarrow> rexp \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
1136 |
where
|
|
|
1137 |
"PC32 s r r' \<equiv> \<forall>y. s \<notin> (L r ;; {y})"
|
|
|
1138 |
|
|
|
1139 |
definition
|
|
|
1140 |
PC42 :: "string \<Rightarrow> string \<Rightarrow> rexp \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
1141 |
where
|
|
|
1142 |
"PC42 s s' r r' \<equiv> (\<forall>x. (s @ x \<in> L r \<longrightarrow> (\<exists>y. s' \<in> {x} ;; (L r' ;; {y})) \<longrightarrow> x = []))"
|
|
|
1143 |
|
|
|
1144 |
|
|
|
1145 |
lemma
|
|
|
1146 |
L1: "\<not>(nullable r1) \<longrightarrow> [] \<in> L r2 \<longrightarrow> PC32 [] r1 r2" and
|
|
|
1147 |
L2: "s1 \<in> L(r1) \<longrightarrow> [] \<in> L(r2) \<longrightarrow> PC42 s1 [] r1 r2" and
|
|
|
1148 |
L3: "s2 \<in> L(der c r2) \<longrightarrow> PC32 s2 (der c r1) (der c r2) \<longrightarrow> PC32 (c#s2) r1 r2" and
|
|
|
1149 |
L4: "s1 \<in> L(der c r1) \<longrightarrow> s2 \<in> L(r2) \<longrightarrow> PC42 s1 s2 (der c r1) r2 \<longrightarrow> PC42 (c#s1) s2 r1 r2" and
|
|
|
1150 |
L5: "nullable(r1) \<longrightarrow> s2 \<in> L(der c r2) \<longrightarrow> PC32 s2 (SEQ (der c r1) r2) (der c r2) \<longrightarrow> PC42 [] (c#s2) r1 r2" and
|
|
|
1151 |
L6: "s0 \<in> L(der c r0) \<longrightarrow> s \<in> L(STAR r0) \<longrightarrow> PC42 s0 s (der c r0) (STAR r0) \<longrightarrow> PC42 (c#s0) s r0 (STAR r0)" and
|
|
|
1152 |
L7: "s' \<in> L(r') \<longrightarrow> s' \<in> L(r) \<longrightarrow> \<not>PC32 s' r r'" and
|
|
|
1153 |
L8: "s \<in> L(r) \<longrightarrow> s' \<in> L(r') \<longrightarrow> s @ x \<in> L(r) \<longrightarrow> s' \<in> {x} ;; (L(r') ;; {y}) \<longrightarrow> x \<noteq> [] \<longrightarrow> \<not>PC42 s s' r r'"
|
|
|
1154 |
apply(auto simp add: PC32_def PC42_def)[1]
|
|
|
1155 |
apply(simp add: Sequ_def)
|
|
|
1156 |
apply (metis nullable_correctness)
|
|
|
1157 |
apply(auto simp add: PC32_def PC42_def Sequ_def)[1]
|
|
|
1158 |
apply(auto simp add: PC32_def PC42_def Sequ_def der_correctness Der_def)[1]
|
|
|
1159 |
apply(simp add: Cons_eq_append_conv)
|
|
|
1160 |
apply(auto)[1]
|
|
|
1161 |
defer
|
|
|
1162 |
apply(auto simp add: PC32_def PC42_def Sequ_def der_correctness Der_def)[1]
|
|
|
1163 |
apply(auto simp add: PC32_def PC42_def Sequ_def der_correctness Der_def nullable_correctness)[1]
|
|
|
1164 |
apply (metis append_Cons append_assoc hd_Cons_tl list.discI list.inject)
|
|
|
1165 |
apply(auto simp add: PC32_def PC42_def Sequ_def der_correctness Der_def)[1]
|
|
|
1166 |
apply(auto simp add: PC32_def PC42_def Sequ_def der_correctness Der_def)[1]
|
|
|
1167 |
apply(auto simp add: PC32_def PC42_def Sequ_def der_correctness Der_def)[1]
|
|
|
1168 |
oops
|
|
|
1169 |
|
|
|
1170 |
definition
|
|
|
1171 |
PC33 :: "string \<Rightarrow> rexp \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
1172 |
where
|
|
|
1173 |
"PC33 s r r' \<equiv> s \<notin> L r"
|
|
|
1174 |
|
|
|
1175 |
definition
|
|
|
1176 |
PC43 :: "string \<Rightarrow> string \<Rightarrow> rexp \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
1177 |
where
|
|
|
1178 |
"PC43 s s' r r' \<equiv> (\<forall>x. (s @ x \<in> L r \<longrightarrow> (\<exists>y. s' \<in> {x} ;; (L r' ;; {y})) \<longrightarrow> x = []))"
|
|
|
1179 |
|
|
|
1180 |
lemma
|
|
|
1181 |
L1: "\<not>(nullable r1) \<longrightarrow> [] \<in> L r2 \<longrightarrow> PC33 [] r1 r2" and
|
|
|
1182 |
L2: "s1 \<in> L(r1) \<longrightarrow> [] \<in> L(r2) \<longrightarrow> PC43 s1 [] r1 r2" and
|
|
|
1183 |
L3: "s2 \<in> L(der c r2) \<longrightarrow> PC33 s2 (der c r1) (der c r2) \<longrightarrow> PC33 (c#s2) r1 r2" and
|
|
|
1184 |
L4: "s1 \<in> L(der c r1) \<longrightarrow> s2 \<in> L(r2) \<longrightarrow> PC43 s1 s2 (der c r1) r2 \<longrightarrow> PC43 (c#s1) s2 r1 r2" and
|
|
|
1185 |
L5: "nullable(r1) \<longrightarrow> s2 \<in> L(der c r2) \<longrightarrow> PC33 s2 (SEQ (der c r1) r2) (der c r2) \<longrightarrow> PC43 [] (c#s2) r1 r2" and
|
|
|
1186 |
L6: "s0 \<in> L(der c r0) \<longrightarrow> s \<in> L(STAR r0) \<longrightarrow> PC43 s0 s (der c r0) (STAR r0) \<longrightarrow> PC43 (c#s0) s r0 (STAR r0)" and
|
|
|
1187 |
L7: "s' \<in> L(r') \<longrightarrow> s' \<in> L(r) \<longrightarrow> \<not>PC33 s' r r'" and
|
|
|
1188 |
L8: "s \<in> L(r) \<longrightarrow> s' \<in> L(r') \<longrightarrow> s @ x \<in> L(r) \<longrightarrow> s' \<in> {x} ;; (L(r') ;; {y}) \<longrightarrow> x \<noteq> [] \<longrightarrow> \<not>PC43 s s' r r'"
|
|
|
1189 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1190 |
apply (metis nullable_correctness)
|
|
|
1191 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1192 |
apply(simp add: Sequ_def)
|
|
|
1193 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1194 |
apply(simp add: der_correctness Der_def)
|
|
|
1195 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1196 |
apply(simp add: der_correctness Der_def Sequ_def)
|
|
|
1197 |
apply metis
|
|
|
1198 |
(* 5 *)
|
|
|
1199 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1200 |
apply(simp add: Sequ_def)
|
|
|
1201 |
apply(simp add: der_correctness Der_def)
|
|
|
1202 |
apply(auto)[1]
|
|
|
1203 |
defer
|
|
|
1204 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1205 |
apply(simp add: Sequ_def)
|
|
|
1206 |
apply(simp add: der_correctness Der_def)
|
|
|
1207 |
apply metis
|
|
|
1208 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1209 |
apply(auto simp add: PC33_def PC43_def)[1]
|
|
|
1210 |
(* 5 fails *)
|
|
|
1211 |
apply(simp add: Cons_eq_append_conv)
|
|
|
1212 |
apply(auto)[1]
|
|
|
1213 |
apply(drule_tac x="ys'" in spec)
|
|
|
1214 |
apply(simp)
|
|
|
1215 |
oops
|
|
|
1216 |
|
|
|
1217 |
section {* Roy's Definition *}
|
|
|
1218 |
|
|
|
1219 |
inductive
|
|
|
1220 |
Roy :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<rhd> _ : _" [100, 100] 100)
|
|
|
1221 |
where
|
|
|
1222 |
"\<rhd> Void : EMPTY"
|
|
|
1223 |
| "\<rhd> Char c : CHAR c"
|
|
|
1224 |
| "\<rhd> v : r1 \<Longrightarrow> \<rhd> Left v : ALT r1 r2"
|
|
|
1225 |
| "\<lbrakk>\<rhd> v : r2; flat v \<notin> L r1\<rbrakk> \<Longrightarrow> \<rhd> Right v : ALT r1 r2"
|
|
|
1226 |
| "\<lbrakk>\<rhd> v1 : r1; \<rhd> v2 : r2; \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = flat v2 \<and> (flat v1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)\<rbrakk> \<Longrightarrow>
|
|
|
1227 |
\<rhd> Seq v1 v2 : SEQ r1 r2"
|
|
|
1228 |
| "\<lbrakk>\<rhd> v : r; \<rhd> Stars vs : STAR r; flat v \<noteq> [];
|
|
|
1229 |
\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = flat (Stars vs) \<and> (flat v @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk> \<Longrightarrow>
|
|
|
1230 |
\<rhd> Stars (v#vs) : STAR r"
|
|
|
1231 |
| "\<rhd> Stars [] : STAR r"
|
|
|
1232 |
|
|
|
1233 |
lemma drop_append:
|
|
|
1234 |
assumes "s1 \<sqsubseteq> s2"
|
|
|
1235 |
shows "s1 @ drop (length s1) s2 = s2"
|
|
|
1236 |
using assms
|
|
|
1237 |
apply(simp add: prefix_def)
|
|
|
1238 |
apply(auto)
|
|
|
1239 |
done
|
|
|
1240 |
|
|
|
1241 |
lemma royA:
|
|
|
1242 |
assumes "\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = flat v2 \<and> (flat v1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)"
|
|
|
1243 |
shows "\<forall>s. (s \<in> L(ders (flat v1) r1) \<and>
|
|
|
1244 |
s \<sqsubseteq> (flat v2) \<and> drop (length s) (flat v2) \<in> L r2 \<longrightarrow> s = [])"
|
|
|
1245 |
using assms
|
|
|
1246 |
apply -
|
|
|
1247 |
apply(rule allI)
|
|
|
1248 |
apply(rule impI)
|
|
|
1249 |
apply(simp add: ders_correctness)
|
|
|
1250 |
apply(simp add: Ders_def)
|
|
|
1251 |
thm rest_def
|
|
|
1252 |
apply(drule_tac x="s" in spec)
|
|
|
1253 |
apply(simp)
|
|
|
1254 |
apply(erule disjE)
|
|
|
1255 |
apply(simp)
|
|
|
1256 |
apply(drule_tac x="drop (length s) (flat v2)" in spec)
|
|
|
1257 |
apply(simp add: drop_append)
|
|
|
1258 |
done
|
|
|
1259 |
|
|
|
1260 |
lemma royB:
|
|
|
1261 |
assumes "\<forall>s. (s \<in> L(ders (flat v1) r1) \<and>
|
|
|
1262 |
s \<sqsubseteq> (flat v2) \<and> drop (length s) (flat v2) \<in> L r2 \<longrightarrow> s = [])"
|
|
|
1263 |
shows "\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = flat v2 \<and> (flat v1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)"
|
|
|
1264 |
using assms
|
|
|
1265 |
apply -
|
|
|
1266 |
apply(auto simp add: prefix_def ders_correctness Ders_def)
|
|
|
1267 |
by (metis append_eq_conv_conj)
|
|
|
1268 |
|
|
|
1269 |
lemma royC:
|
|
|
1270 |
assumes "\<forall>s t. (s \<in> L(ders (flat v1) r1) \<and>
|
|
|
1271 |
s \<sqsubseteq> (flat v2 @ t) \<and> drop (length s) (flat v2 @ t) \<in> L r2 \<longrightarrow> s = [])"
|
|
|
1272 |
shows "\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = flat v2 \<and> (flat v1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)"
|
|
|
1273 |
using assms
|
|
|
1274 |
apply -
|
|
|
1275 |
apply(rule royB)
|
|
|
1276 |
apply(rule allI)
|
|
|
1277 |
apply(drule_tac x="s" in spec)
|
|
|
1278 |
apply(drule_tac x="[]" in spec)
|
|
|
1279 |
apply(simp)
|
|
|
1280 |
done
|
|
|
1281 |
|
|
|
1282 |
inductive
|
|
|
1283 |
Roy2 :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("2\<rhd> _ : _" [100, 100] 100)
|
|
|
1284 |
where
|
|
|
1285 |
"2\<rhd> Void : EMPTY"
|
|
|
1286 |
| "2\<rhd> Char c : CHAR c"
|
|
|
1287 |
| "2\<rhd> v : r1 \<Longrightarrow> 2\<rhd> Left v : ALT r1 r2"
|
|
|
1288 |
| "\<lbrakk>2\<rhd> v : r2; \<forall>t. flat v \<notin> (L r1 ;; {t})\<rbrakk> \<Longrightarrow> 2\<rhd> Right v : ALT r1 r2"
|
|
|
1289 |
| "\<lbrakk>2\<rhd> v1 : r1; 2\<rhd> v2 : r2;
|
|
|
1290 |
\<forall>s. ((flat v1 @ s \<in> L r1) \<and>
|
|
|
1291 |
(\<exists>t. s \<sqsubseteq> (flat v2 @ t) \<and> drop (length s) (flat v2) \<in> (L r2 ;; {t}))) \<longrightarrow> s = []\<rbrakk> \<Longrightarrow>
|
|
|
1292 |
2\<rhd> Seq v1 v2 : SEQ r1 r2"
|
|
|
1293 |
| "\<lbrakk>2\<rhd> v : r; 2\<rhd> Stars vs : STAR r; flat v \<noteq> [];
|
|
|
1294 |
\<forall>s. ((flat v @ s \<in> L r) \<and>
|
|
|
1295 |
(\<exists>t. s \<sqsubseteq> (flat (Stars vs) @ t) \<and> drop (length s) (flat (Stars vs)) \<in> (L (STAR r) ;; {t}))) \<longrightarrow> s = []\<rbrakk>
|
|
|
1296 |
\<Longrightarrow> 2\<rhd> Stars (v#vs) : STAR r"
|
|
|
1297 |
| "2\<rhd> Stars [] : STAR r"
|
|
|
1298 |
|
|
|
1299 |
lemma Roy2_props:
|
|
|
1300 |
assumes "2\<rhd> v : r"
|
|
|
1301 |
shows "\<turnstile> v : r"
|
|
|
1302 |
using assms
|
|
|
1303 |
apply(induct)
|
|
|
1304 |
apply(auto intro: Prf.intros)
|
|
|
1305 |
done
|
|
|
1306 |
|
|
|
1307 |
lemma Roy_mkeps_nullable:
|
|
|
1308 |
assumes "nullable(r)"
|
|
|
1309 |
shows "2\<rhd> (mkeps r) : r"
|
|
|
1310 |
using assms
|
|
|
1311 |
apply(induct rule: nullable.induct)
|
|
|
1312 |
apply(auto intro: Roy2.intros)
|
|
|
1313 |
apply(rule Roy2.intros)
|
|
|
1314 |
apply(simp_all)
|
|
|
1315 |
apply(simp add: mkeps_flat)
|
|
|
1316 |
apply(simp add: Sequ_def)
|
|
|
1317 |
apply (metis nullable_correctness)
|
|
|
1318 |
apply(rule Roy2.intros)
|
|
|
1319 |
apply(simp_all)
|
|
|
1320 |
apply(rule allI)
|
|
|
1321 |
apply(rule impI)
|
|
|
1322 |
apply(auto simp add: Sequ_def)
|
|
|
1323 |
apply(simp add: mkeps_flat)
|
|
|
1324 |
apply(auto simp add: prefix_def)
|
|
|
1325 |
done
|
|
|
1326 |
|
|
|
1327 |
section {* Alternative Posix definition *}
|
|
|
1328 |
|
|
|
1329 |
inductive
|
|
|
1330 |
PMatch :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
|
|
|
1331 |
where
|
|
|
1332 |
"[] \<in> EMPTY \<rightarrow> Void"
|
|
|
1333 |
| "[c] \<in> (CHAR c) \<rightarrow> (Char c)"
|
|
|
1334 |
| "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
|
|
|
1335 |
| "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
|
|
|
1336 |
| "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
|
|
|
1337 |
\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r1 \<and> s\<^sub>4 \<in> L r2)\<rbrakk> \<Longrightarrow>
|
|
|
1338 |
(s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"
|
|
|
1339 |
| "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];
|
|
|
1340 |
\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
|
|
|
1341 |
\<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"
|
|
|
1342 |
| "[] \<in> STAR r \<rightarrow> Stars []"
|
|
|
1343 |
|
|
|
1344 |
inductive
|
|
|
1345 |
PMatchX :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("\<turnstile> _ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
|
|
|
1346 |
where
|
|
|
1347 |
"\<turnstile> s \<in> EMPTY \<rightarrow> Void"
|
|
|
1348 |
| "\<turnstile> (c # s) \<in> (CHAR c) \<rightarrow> (Char c)"
|
|
|
1349 |
| "\<turnstile> s \<in> r1 \<rightarrow> v \<Longrightarrow> \<turnstile> s \<in> (ALT r1 r2) \<rightarrow> (Left v)"
|
|
|
1350 |
| "\<lbrakk>\<turnstile> s \<in> r2 \<rightarrow> v; \<not>(\<exists>s'. s' \<sqsubseteq> s \<and> flat v \<sqsubseteq> s' \<and> s' \<in> L(r1))\<rbrakk> \<Longrightarrow> \<turnstile> s \<in> (ALT r1 r2) \<rightarrow> (Right v)"
|
|
|
1351 |
| "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; \<turnstile> s2 \<in> r2 \<rightarrow> v2;
|
|
|
1352 |
\<not>(\<exists>s3 s4. s3 \<noteq> [] \<and> (s3 @ s4) \<sqsubseteq> s2 \<and> (s1 @ s3) \<in> L r1 \<and> s4 \<in> L r2)\<rbrakk> \<Longrightarrow>
|
|
|
1353 |
\<turnstile> (s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)"
|
|
|
1354 |
| "\<lbrakk>s1 \<in> r \<rightarrow> v; \<turnstile> s2 \<in> STAR r \<rightarrow> Stars vs; flat v \<noteq> [];
|
|
|
1355 |
\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> (s\<^sub>3 @ s\<^sub>4) \<sqsubseteq> s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
|
|
|
1356 |
\<Longrightarrow> \<turnstile> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"
|
|
|
1357 |
| "\<turnstile> s \<in> STAR r \<rightarrow> Stars []"
|
|
|
1358 |
|
|
|
1359 |
lemma PMatch1:
|
|
|
1360 |
assumes "s \<in> r \<rightarrow> v"
|
|
|
1361 |
shows "\<turnstile> v : r" "flat v = s"
|
|
|
1362 |
using assms
|
|
|
1363 |
apply(induct s r v rule: PMatch.induct)
|
|
|
1364 |
apply(auto)
|
|
|
1365 |
apply (metis Prf.intros(4))
|
|
|
1366 |
apply (metis Prf.intros(5))
|
|
|
1367 |
apply (metis Prf.intros(2))
|
|
|
1368 |
apply (metis Prf.intros(3))
|
|
|
1369 |
apply (metis Prf.intros(1))
|
|
|
1370 |
apply (metis Prf.intros(7))
|
|
|
1371 |
by (metis Prf.intros(6))
|
|
|
1372 |
|
|
|
1373 |
|
|
|
1374 |
lemma PMatchX1:
|
|
|
1375 |
assumes "\<turnstile> s \<in> r \<rightarrow> v"
|
|
|
1376 |
shows "\<turnstile> v : r"
|
|
|
1377 |
using assms
|
|
|
1378 |
apply(induct s r v rule: PMatchX.induct)
|
|
|
1379 |
apply(auto simp add: prefix_def intro: Prf.intros)
|
|
|
1380 |
apply (metis PMatch1(1) Prf.intros(1))
|
|
|
1381 |
by (metis PMatch1(1) Prf.intros(7))
|
|
|
1382 |
|
|
|
1383 |
|
|
|
1384 |
lemma PMatchX:
|
|
|
1385 |
assumes "\<turnstile> s \<in> r \<rightarrow> v"
|
|
|
1386 |
shows "flat v \<sqsubseteq> s"
|
|
|
1387 |
using assms
|
|
|
1388 |
apply(induct s r v rule: PMatchX.induct)
|
|
|
1389 |
apply(auto simp add: prefix_def PMatch1)
|
|
|
1390 |
done
|
|
|
1391 |
|
|
|
1392 |
lemma PMatchX_PMatch:
|
|
|
1393 |
assumes "\<turnstile> s \<in> r \<rightarrow> v" "flat v = s"
|
|
|
1394 |
shows "s \<in> r \<rightarrow> v"
|
|
|
1395 |
using assms
|
|
|
1396 |
apply(induct s r v rule: PMatchX.induct)
|
|
|
1397 |
apply(auto intro: PMatch.intros)
|
|
|
1398 |
apply(rule PMatch.intros)
|
|
|
1399 |
apply(simp)
|
|
|
1400 |
apply (metis PMatchX Prefixes_def mem_Collect_eq)
|
|
|
1401 |
apply (smt2 PMatch.intros(5) PMatch1(2) PMatchX append_Nil2 append_assoc append_self_conv prefix_def)
|
|
|
1402 |
by (metis L.simps(6) PMatch.intros(6) PMatch1(2) append_Nil2 append_eq_conv_conj prefix_def)
|
|
|
1403 |
|
|
|
1404 |
lemma PMatch_PMatchX:
|
|
|
1405 |
assumes "s \<in> r \<rightarrow> v"
|
|
|
1406 |
shows "\<turnstile> s \<in> r \<rightarrow> v"
|
|
|
1407 |
using assms
|
|
|
1408 |
apply(induct s r v arbitrary: s' rule: PMatch.induct)
|
|
|
1409 |
apply(auto intro: PMatchX.intros)
|
|
|
1410 |
apply(rule PMatchX.intros)
|
|
|
1411 |
apply(simp)
|
|
|
1412 |
apply(rule notI)
|
|
|
1413 |
apply(auto)[1]
|
|
|
1414 |
apply (metis PMatch1(2) append_eq_conv_conj length_sprefix less_imp_le_nat prefix_def sprefix_def take_all)
|
|
|
1415 |
apply(rule PMatchX.intros)
|
|
|
1416 |
apply(simp)
|
|
|
1417 |
apply(simp)
|
|
|
1418 |
apply(auto)[1]
|
|
|
1419 |
oops
|
|
|
1420 |
|
|
|
1421 |
lemma
|
|
|
1422 |
assumes "\<rhd> v : r"
|
|
|
1423 |
shows "(flat v) \<in> r \<rightarrow> v"
|
|
|
1424 |
using assms
|
|
|
1425 |
apply(induct)
|
|
|
1426 |
apply(auto intro: PMatch.intros)
|
|
|
1427 |
apply(rule PMatch.intros)
|
|
|
1428 |
apply(simp)
|
|
|
1429 |
apply(simp)
|
|
|
1430 |
apply(simp)
|
|
|
1431 |
apply(auto)[1]
|
|
|
1432 |
done
|
|
|
1433 |
|
|
|
1434 |
lemma
|
|
|
1435 |
assumes "s \<in> r \<rightarrow> v"
|
|
|
1436 |
shows "\<rhd> v : r"
|
|
|
1437 |
using assms
|
|
|
1438 |
apply(induct)
|
|
|
1439 |
apply(auto intro: Roy.intros)
|
|
|
1440 |
apply (metis PMatch1(2) Roy.intros(4))
|
|
|
1441 |
apply (metis PMatch1(2) Roy.intros(5))
|
|
|
1442 |
by (metis L.simps(6) PMatch1(2) Roy.intros(6))
|
|
|
1443 |
|
|
|
1444 |
|
|
|
1445 |
lemma PMatch_mkeps:
|
|
|
1446 |
assumes "nullable r"
|
|
|
1447 |
shows "[] \<in> r \<rightarrow> mkeps r"
|
|
|
1448 |
using assms
|
|
|
1449 |
apply(induct r)
|
|
|
1450 |
apply(auto)
|
|
|
1451 |
apply (metis PMatch.intros(1))
|
|
|
1452 |
apply(subst append.simps(1)[symmetric])
|
|
|
1453 |
apply (rule PMatch.intros)
|
|
|
1454 |
apply(simp)
|
|
|
1455 |
apply(simp)
|
|
|
1456 |
apply(auto)[1]
|
|
|
1457 |
apply (rule PMatch.intros)
|
|
|
1458 |
apply(simp)
|
|
|
1459 |
apply (rule PMatch.intros)
|
|
|
1460 |
apply(simp)
|
|
|
1461 |
apply (rule PMatch.intros)
|
|
|
1462 |
apply(simp)
|
|
|
1463 |
apply (metis nullable_correctness)
|
|
|
1464 |
apply(metis PMatch.intros(7))
|
|
|
1465 |
done
|
|
|
1466 |
|
|
|
1467 |
|
|
|
1468 |
lemma PMatch1N:
|
|
|
1469 |
assumes "s \<in> r \<rightarrow> v"
|
|
|
1470 |
shows "\<Turnstile> v : r"
|
|
|
1471 |
using assms
|
|
|
1472 |
apply(induct s r v rule: PMatch.induct)
|
|
|
1473 |
apply(auto)
|
|
|
1474 |
apply (metis NPrf.intros(4))
|
|
|
1475 |
apply (metis NPrf.intros(5))
|
|
|
1476 |
apply (metis NPrf.intros(2))
|
|
|
1477 |
apply (metis NPrf.intros(3))
|
|
|
1478 |
apply (metis NPrf.intros(1))
|
|
|
1479 |
apply(rule NPrf.intros)
|
|
|
1480 |
apply(simp)
|
|
|
1481 |
apply(simp)
|
|
|
1482 |
apply(simp)
|
|
|
1483 |
apply(rule NPrf.intros)
|
|
|
1484 |
done
|
|
|
1485 |
|
|
|
1486 |
lemma PMatch_determ:
|
|
|
1487 |
shows "\<lbrakk>s \<in> r \<rightarrow> v1; s \<in> r \<rightarrow> v2\<rbrakk> \<Longrightarrow> v1 = v2"
|
|
|
1488 |
and "\<lbrakk>s \<in> (STAR r) \<rightarrow> Stars vs1; s \<in> (STAR r) \<rightarrow> Stars vs2\<rbrakk> \<Longrightarrow> vs1 = vs2"
|
|
|
1489 |
apply(induct v1 and vs1 arbitrary: s r v2 and s r vs2 rule: val.inducts)
|
|
|
1490 |
apply(erule PMatch.cases)
|
|
|
1491 |
apply(simp_all)[7]
|
|
|
1492 |
apply(erule PMatch.cases)
|
|
|
1493 |
apply(simp_all)[7]
|
|
|
1494 |
apply(erule PMatch.cases)
|
|
|
1495 |
apply(simp_all)[7]
|
|
|
1496 |
apply(erule PMatch.cases)
|
|
|
1497 |
apply(simp_all)[7]
|
|
|
1498 |
apply(erule PMatch.cases)
|
|
|
1499 |
apply(simp_all)[7]
|
|
|
1500 |
apply(erule PMatch.cases)
|
|
|
1501 |
apply(simp_all)[7]
|
|
|
1502 |
apply(clarify)
|
|
|
1503 |
apply(subgoal_tac "s1 = s1a \<and> s2 = s2a")
|
|
|
1504 |
apply metis
|
|
|
1505 |
apply(rule conjI)
|
|
|
1506 |
apply(simp add: append_eq_append_conv2)
|
|
|
1507 |
apply(auto)[1]
|
|
|
1508 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1509 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1510 |
apply(simp add: append_eq_append_conv2)
|
|
|
1511 |
apply(auto)[1]
|
|
|
1512 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1513 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1514 |
apply(erule PMatch.cases)
|
|
|
1515 |
apply(simp_all)[7]
|
|
|
1516 |
apply(clarify)
|
|
|
1517 |
apply(erule PMatch.cases)
|
|
|
1518 |
apply(simp_all)[7]
|
|
|
1519 |
apply(clarify)
|
|
|
1520 |
apply (metis NPrf_flat_L PMatch1(2) PMatch1N)
|
|
|
1521 |
apply(erule PMatch.cases)
|
|
|
1522 |
apply(simp_all)[7]
|
|
|
1523 |
apply(clarify)
|
|
|
1524 |
apply(erule PMatch.cases)
|
|
|
1525 |
apply(simp_all)[7]
|
|
|
1526 |
apply(clarify)
|
|
|
1527 |
apply (metis NPrf_flat_L PMatch1(2) PMatch1N)
|
|
|
1528 |
(* star case *)
|
|
|
1529 |
defer
|
|
|
1530 |
apply(erule PMatch.cases)
|
|
|
1531 |
apply(simp_all)[7]
|
|
|
1532 |
apply(clarify)
|
|
|
1533 |
apply(erule PMatch.cases)
|
|
|
1534 |
apply(simp_all)[7]
|
|
|
1535 |
apply(clarify)
|
|
|
1536 |
apply (metis PMatch1(2))
|
|
|
1537 |
apply(rotate_tac 3)
|
|
|
1538 |
apply(erule PMatch.cases)
|
|
|
1539 |
apply(simp_all)[7]
|
|
|
1540 |
apply(clarify)
|
|
|
1541 |
apply(erule PMatch.cases)
|
|
|
1542 |
apply(simp_all)[7]
|
|
|
1543 |
apply(clarify)
|
|
|
1544 |
apply(subgoal_tac "s1 = s1a \<and> s2 = s2a")
|
|
|
1545 |
apply metis
|
|
|
1546 |
apply(simp add: append_eq_append_conv2)
|
|
|
1547 |
apply(auto)[1]
|
|
|
1548 |
apply (metis L.simps(6) PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1549 |
apply (metis L.simps(6) PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1550 |
apply (metis L.simps(6) PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1551 |
apply (metis L.simps(6) PMatch1(1) PMatch1(2) Prf_flat_L)
|
|
|
1552 |
apply(erule PMatch.cases)
|
|
|
1553 |
apply(simp_all)[7]
|
|
|
1554 |
apply(clarify)
|
|
|
1555 |
apply (metis PMatch1(2))
|
|
|
1556 |
apply(erule PMatch.cases)
|
|
|
1557 |
apply(simp_all)[7]
|
|
|
1558 |
apply(clarify)
|
|
|
1559 |
apply(erule PMatch.cases)
|
|
|
1560 |
apply(simp_all)[7]
|
|
|
1561 |
apply(clarify)
|
|
|
1562 |
apply(subgoal_tac "s1 = s1a \<and> s2 = s2a")
|
|
|
1563 |
apply(drule_tac x="s1 @ s2" in meta_spec)
|
|
|
1564 |
apply(drule_tac x="rb" in meta_spec)
|
|
|
1565 |
apply(drule_tac x="(va#vsa)" in meta_spec)
|
|
|
1566 |
apply(simp)
|
|
|
1567 |
apply(drule meta_mp)
|
|
|
1568 |
apply (metis L.simps(6) PMatch.intros(6))
|
|
|
1569 |
apply (metis L.simps(6) PMatch.intros(6))
|
|
|
1570 |
apply(simp add: append_eq_append_conv2)
|
|
|
1571 |
apply(auto)[1]
|
|
|
1572 |
apply (metis L.simps(6) NPrf_flat_L PMatch1(2) PMatch1N)
|
|
|
1573 |
apply (metis L.simps(6) NPrf_flat_L PMatch1(2) PMatch1N)
|
|
|
1574 |
apply (metis L.simps(6) NPrf_flat_L PMatch1(2) PMatch1N)
|
|
|
1575 |
apply (metis L.simps(6) NPrf_flat_L PMatch1(2) PMatch1N)
|
|
|
1576 |
apply (metis PMatch1(2))
|
|
|
1577 |
apply(erule PMatch.cases)
|
|
|
1578 |
apply(simp_all)[7]
|
|
|
1579 |
apply(clarify)
|
|
|
1580 |
by (metis PMatch1(2))
|
|
|
1581 |
|
|
|
1582 |
|
|
|
1583 |
lemma PMatch_Values:
|
|
|
1584 |
assumes "s \<in> r \<rightarrow> v"
|
|
|
1585 |
shows "v \<in> Values r s"
|
|
|
1586 |
using assms
|
|
|
1587 |
apply(simp add: Values_def PMatch1)
|
|
|
1588 |
by (metis append_Nil2 prefix_def)
|
|
|
1589 |
|
|
|
1590 |
lemma PMatch2:
|
|
|
1591 |
assumes "s \<in> (der c r) \<rightarrow> v"
|
|
|
1592 |
shows "(c#s) \<in> r \<rightarrow> (injval r c v)"
|
|
|
1593 |
using assms
|
|
|
1594 |
apply(induct c r arbitrary: s v rule: der.induct)
|
|
|
1595 |
apply(auto)
|
|
|
1596 |
apply(erule PMatch.cases)
|
|
|
1597 |
apply(simp_all)[7]
|
|
|
1598 |
apply(erule PMatch.cases)
|
|
|
1599 |
apply(simp_all)[7]
|
|
|
1600 |
apply(case_tac "c = c'")
|
|
|
1601 |
apply(simp)
|
|
|
1602 |
apply(erule PMatch.cases)
|
|
|
1603 |
apply(simp_all)[7]
|
|
|
1604 |
apply (metis PMatch.intros(2))
|
|
|
1605 |
apply(simp)
|
|
|
1606 |
apply(erule PMatch.cases)
|
|
|
1607 |
apply(simp_all)[7]
|
|
|
1608 |
apply(erule PMatch.cases)
|
|
|
1609 |
apply(simp_all)[7]
|
|
|
1610 |
apply (metis PMatch.intros(3))
|
|
|
1611 |
apply(clarify)
|
|
|
1612 |
apply(rule PMatch.intros)
|
|
|
1613 |
apply metis
|
|
|
1614 |
apply(simp add: L_flat_NPrf)
|
|
|
1615 |
apply(auto)[1]
|
|
|
1616 |
apply(frule_tac c="c" in v3_proj)
|
|
|
1617 |
apply metis
|
|
|
1618 |
apply(drule_tac x="projval r1 c v" in spec)
|
|
|
1619 |
apply(drule mp)
|
|
|
1620 |
apply (metis v4_proj2)
|
|
|
1621 |
apply (metis NPrf_imp_Prf)
|
|
|
1622 |
(* SEQ case *)
|
|
|
1623 |
apply(case_tac "nullable r1")
|
|
|
1624 |
apply(simp)
|
|
|
1625 |
prefer 2
|
|
|
1626 |
apply(simp)
|
|
|
1627 |
apply(erule PMatch.cases)
|
|
|
1628 |
apply(simp_all)[7]
|
|
|
1629 |
apply(clarify)
|
|
|
1630 |
apply(subst append.simps(2)[symmetric])
|
|
|
1631 |
apply(rule PMatch.intros)
|
|
|
1632 |
apply metis
|
|
|
1633 |
apply metis
|
|
|
1634 |
apply(auto)[1]
|
|
|
1635 |
apply(simp add: der_correctness Der_def)
|
|
|
1636 |
apply(auto)[1]
|
|
|
1637 |
(* nullable case *)
|
|
|
1638 |
apply(erule PMatch.cases)
|
|
|
1639 |
apply(simp_all)[7]
|
|
|
1640 |
apply(clarify)
|
|
|
1641 |
apply(erule PMatch.cases)
|
|
|
1642 |
apply(simp_all)[4]
|
|
|
1643 |
apply(clarify)
|
|
|
1644 |
apply(simp (no_asm))
|
|
|
1645 |
apply(subst append.simps(2)[symmetric])
|
|
|
1646 |
apply(rule PMatch.intros)
|
|
|
1647 |
apply metis
|
|
|
1648 |
apply metis
|
|
|
1649 |
apply(erule contrapos_nn)
|
|
|
1650 |
apply(erule exE)+
|
|
|
1651 |
apply(auto)[1]
|
|
|
1652 |
apply(simp add: L_flat_NPrf)
|
|
|
1653 |
apply(auto)[1]
|
|
|
1654 |
thm v3_proj
|
|
|
1655 |
apply(frule_tac c="c" in v3_proj)
|
|
|
1656 |
apply metis
|
|
|
1657 |
apply(rule_tac x="s\<^sub>3" in exI)
|
|
|
1658 |
apply(simp)
|
|
|
1659 |
apply (metis NPrf_imp_Prf v4_proj2)
|
|
|
1660 |
apply(simp)
|
|
|
1661 |
(* interesting case *)
|
|
|
1662 |
apply(clarify)
|
|
|
1663 |
apply(clarify)
|
|
|
1664 |
apply(simp)
|
|
|
1665 |
apply(subst (asm) L.simps(4)[symmetric])
|
|
|
1666 |
apply(simp only: L_flat_Prf)
|
|
|
1667 |
apply(simp)
|
|
|
1668 |
apply(subst append.simps(1)[symmetric])
|
|
|
1669 |
apply(rule PMatch.intros)
|
|
|
1670 |
apply (metis PMatch_mkeps)
|
|
|
1671 |
apply metis
|
|
|
1672 |
apply(auto)
|
|
|
1673 |
apply(simp only: L_flat_NPrf)
|
|
|
1674 |
apply(simp)
|
|
|
1675 |
apply(auto)
|
|
|
1676 |
apply(drule_tac x="Seq (projval r1 c v) vb" in spec)
|
|
|
1677 |
apply(drule mp)
|
|
|
1678 |
apply(simp)
|
|
|
1679 |
|
|
|
1680 |
apply (metis append_Cons butlast_snoc list.sel(1) neq_Nil_conv rotate1.simps(2) v4_proj2)
|
|
|
1681 |
apply(subgoal_tac "\<turnstile> projval r1 c v : der c r1")
|
|
|
1682 |
apply (metis NPrf_imp_Prf Prf.intros(1))
|
|
|
1683 |
apply(rule NPrf_imp_Prf)
|
|
|
1684 |
apply(rule v3_proj)
|
|
|
1685 |
apply(simp)
|
|
|
1686 |
apply (metis Cons_eq_append_conv)
|
|
|
1687 |
(* Stars case *)
|
|
|
1688 |
apply(erule PMatch.cases)
|
|
|
1689 |
apply(simp_all)[7]
|
|
|
1690 |
apply(clarify)
|
|
|
1691 |
apply(rotate_tac 2)
|
|
|
1692 |
apply(frule_tac PMatch1)
|
|
|
1693 |
apply(erule PMatch.cases)
|
|
|
1694 |
apply(simp_all)[7]
|
|
|
1695 |
apply(subst append.simps(2)[symmetric])
|
|
|
1696 |
apply(rule PMatch.intros)
|
|
|
1697 |
apply metis
|
|
|
1698 |
apply(auto)[1]
|
|
|
1699 |
apply(rule PMatch.intros)
|
|
|
1700 |
apply(simp)
|
|
|
1701 |
apply(simp)
|
|
|
1702 |
apply(simp)
|
|
|
1703 |
apply (metis L.simps(6))
|
|
|
1704 |
apply(subst v4)
|
|
|
1705 |
apply (metis NPrf_imp_Prf PMatch1N)
|
|
|
1706 |
apply(simp)
|
|
|
1707 |
apply(auto)[1]
|
|
|
1708 |
apply(drule_tac x="s\<^sub>3" in spec)
|
|
|
1709 |
apply(drule mp)
|
|
|
1710 |
defer
|
|
|
1711 |
apply metis
|
|
|
1712 |
apply(clarify)
|
|
|
1713 |
apply(drule_tac x="s1" in meta_spec)
|
|
|
1714 |
apply(drule_tac x="v1" in meta_spec)
|
|
|
1715 |
apply(simp)
|
|
|
1716 |
apply(rotate_tac 2)
|
|
|
1717 |
apply(drule PMatch.intros(6))
|
|
|
1718 |
apply(rule PMatch.intros(7))
|
|
|
1719 |
apply (metis PMatch1(1) list.distinct(1) v4)
|
|
|
1720 |
apply (metis Nil_is_append_conv)
|
|
|
1721 |
apply(simp)
|
|
|
1722 |
apply(subst der_correctness)
|
|
|
1723 |
apply(simp add: Der_def)
|
|
|
1724 |
done
|
|
|
1725 |
|
|
|
1726 |
|
|
|
1727 |
|
|
|
1728 |
lemma Sequ_single:
|
|
|
1729 |
"(A ;; {t}) = {s @ t | s . s \<in> A}"
|
|
|
1730 |
apply(simp add: Sequ_def)
|
|
|
1731 |
done
|
|
|
1732 |
|
|
|
1733 |
lemma Sequ_not:
|
|
|
1734 |
assumes "\<forall>t. s \<notin> (L(der c r1) ;; {t})" "L r1 \<noteq> {}"
|
|
|
1735 |
shows "\<forall>t. c # s \<notin> (L r1 ;; {t})"
|
|
|
1736 |
using assms
|
|
|
1737 |
apply(simp add: der_correctness)
|
|
|
1738 |
apply(simp add: Der_def)
|
|
|
1739 |
apply(simp add: Sequ_def)
|
|
|
1740 |
apply(rule allI)+
|
|
|
1741 |
apply(rule impI)
|
|
|
1742 |
apply(simp add: Cons_eq_append_conv)
|
|
|
1743 |
apply(auto)
|
|
|
1744 |
|
|
|
1745 |
oops
|
|
|
1746 |
|
|
|
1747 |
lemma PMatch_Roy2:
|
|
|
1748 |
assumes "2\<rhd> v : (der c r)" "\<exists>s. c # s \<in> L r"
|
|
|
1749 |
shows "2\<rhd> (injval r c v) : r"
|
|
|
1750 |
using assms
|
|
|
1751 |
apply(induct c r arbitrary: v rule: der.induct)
|
|
|
1752 |
apply(auto)
|
|
|
1753 |
apply(erule Roy2.cases)
|
|
|
1754 |
apply(simp_all)
|
|
|
1755 |
apply (metis Roy2.intros(2))
|
|
|
1756 |
(* alt case *)
|
|
|
1757 |
apply(erule Roy2.cases)
|
|
|
1758 |
apply(simp_all)
|
|
|
1759 |
apply(clarify)
|
|
|
1760 |
apply (metis Roy2.intros(3))
|
|
|
1761 |
apply(clarify)
|
|
|
1762 |
apply(rule Roy2.intros(4))
|
|
|
1763 |
apply (metis (full_types) Prf_flat_L Roy2_props v3 v4)
|
|
|
1764 |
apply(subgoal_tac "\<forall>t. c # flat va \<notin> L r1 ;; {t}")
|
|
|
1765 |
prefer 2
|
|
|
1766 |
apply(simp add: der_correctness)
|
|
|
1767 |
apply(simp add: Der_def)
|
|
|
1768 |
apply(simp add: Sequ_def)
|
|
|
1769 |
apply(rule allI)+
|
|
|
1770 |
apply(rule impI)
|
|
|
1771 |
apply(simp add: Cons_eq_append_conv)
|
|
|
1772 |
apply(erule disjE)
|
|
|
1773 |
apply(erule conjE)
|
|
|
1774 |
prefer 2
|
|
|
1775 |
apply metis
|
|
|
1776 |
apply(simp)
|
|
|
1777 |
apply(drule_tac x="[]" in spec)
|
|
|
1778 |
apply(drule_tac x="drop 1 t" in spec)
|
|
|
1779 |
apply(clarify)
|
|
|
1780 |
apply(simp)
|
|
|
1781 |
oops
|
|
|
1782 |
|
|
|
1783 |
lemma lex_correct1:
|
|
|
1784 |
assumes "s \<notin> L r"
|
|
|
1785 |
shows "lex r s = None"
|
|
|
1786 |
using assms
|
|
|
1787 |
apply(induct s arbitrary: r)
|
|
|
1788 |
apply(simp)
|
|
|
1789 |
apply (metis nullable_correctness)
|
|
|
1790 |
apply(auto)
|
|
|
1791 |
apply(drule_tac x="der a r" in meta_spec)
|
|
|
1792 |
apply(drule meta_mp)
|
|
|
1793 |
apply(auto)
|
|
|
1794 |
apply(simp add: L_flat_Prf)
|
|
|
1795 |
by (metis v3 v4)
|
|
|
1796 |
|
|
|
1797 |
|
|
|
1798 |
lemma lex_correct2:
|
|
|
1799 |
assumes "s \<in> L r"
|
|
|
1800 |
shows "\<exists>v. lex r s = Some(v) \<and> \<turnstile> v : r \<and> flat v = s"
|
|
|
1801 |
using assms
|
|
|
1802 |
apply(induct s arbitrary: r)
|
|
|
1803 |
apply(simp)
|
|
|
1804 |
apply (metis mkeps_flat mkeps_nullable nullable_correctness)
|
|
|
1805 |
apply(drule_tac x="der a r" in meta_spec)
|
|
|
1806 |
apply(drule meta_mp)
|
|
|
1807 |
apply(simp add: L_flat_NPrf)
|
|
|
1808 |
apply(auto)
|
|
|
1809 |
apply (metis v3_proj v4_proj2)
|
|
|
1810 |
apply (metis v3)
|
|
|
1811 |
apply(rule v4)
|
|
|
1812 |
by metis
|
|
|
1813 |
|
|
|
1814 |
lemma lex_correct3:
|
|
|
1815 |
assumes "s \<in> L r"
|
|
|
1816 |
shows "\<exists>v. lex r s = Some(v) \<and> s \<in> r \<rightarrow> v"
|
|
|
1817 |
using assms
|
|
|
1818 |
apply(induct s arbitrary: r)
|
|
|
1819 |
apply(simp)
|
|
|
1820 |
apply (metis PMatch_mkeps nullable_correctness)
|
|
|
1821 |
apply(drule_tac x="der a r" in meta_spec)
|
|
|
1822 |
apply(drule meta_mp)
|
|
|
1823 |
apply(simp add: L_flat_NPrf)
|
|
|
1824 |
apply(auto)
|
|
|
1825 |
apply (metis v3_proj v4_proj2)
|
|
|
1826 |
apply(rule PMatch2)
|
|
|
1827 |
apply(simp)
|
|
|
1828 |
done
|
|
|
1829 |
|
|
|
1830 |
lemma lex_correct4:
|
|
|
1831 |
assumes "s \<in> L r"
|
|
|
1832 |
shows "\<exists>v. lex r s = Some(v) \<and> \<Turnstile> v : r \<and> flat v = s"
|
|
|
1833 |
using lex_correct3[OF assms]
|
|
|
1834 |
apply(auto)
|
|
|
1835 |
apply (metis PMatch1N)
|
|
|
1836 |
by (metis PMatch1(2))
|
|
|
1837 |
|
|
|
1838 |
|
|
|
1839 |
lemma lex_correct5:
|
|
|
1840 |
assumes "s \<in> L r"
|
|
|
1841 |
shows "s \<in> r \<rightarrow> (lex2 r s)"
|
|
|
1842 |
using assms
|
|
|
1843 |
apply(induct s arbitrary: r)
|
|
|
1844 |
apply(simp)
|
|
|
1845 |
apply (metis PMatch_mkeps nullable_correctness)
|
|
|
1846 |
apply(simp)
|
|
|
1847 |
apply(rule PMatch2)
|
|
|
1848 |
apply(drule_tac x="der a r" in meta_spec)
|
|
|
1849 |
apply(drule meta_mp)
|
|
|
1850 |
apply(simp add: L_flat_NPrf)
|
|
|
1851 |
apply(auto)
|
|
|
1852 |
apply (metis v3_proj v4_proj2)
|
|
|
1853 |
done
|
|
|
1854 |
|
|
|
1855 |
lemma
|
|
|
1856 |
"lex2 (ALT (CHAR a) (ALT (CHAR b) (SEQ (CHAR a) (CHAR b)))) [a,b] = Right (Right (Seq (Char a) (Char b)))"
|
|
|
1857 |
apply(simp)
|
|
|
1858 |
done
|
|
|
1859 |
|
|
|
1860 |
|
|
|
1861 |
(* NOT DONE YET *)
|
|
|
1862 |
|
|
|
1863 |
section {* Sulzmann's Ordering of values *}
|
|
|
1864 |
|
|
|
1865 |
inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<succ>_ _" [100, 100, 100] 100)
|
|
|
1866 |
where
|
|
|
1867 |
"v2 \<succ>r2 v2' \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1 v2')"
|
|
|
1868 |
| "\<lbrakk>v1 \<succ>r1 v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')"
|
|
|
1869 |
| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Right v2)"
|
|
|
1870 |
| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Left v1)"
|
|
|
1871 |
| "v2 \<succ>r2 v2' \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Right v2')"
|
|
|
1872 |
| "v1 \<succ>r1 v1' \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Left v1')"
|
|
|
1873 |
| "Void \<succ>EMPTY Void"
|
|
|
1874 |
| "(Char c) \<succ>(CHAR c) (Char c)"
|
|
|
1875 |
| "flat (Stars (v # vs)) = [] \<Longrightarrow> (Stars []) \<succ>(STAR r) (Stars (v # vs))"
|
|
|
1876 |
| "flat (Stars (v # vs)) \<noteq> [] \<Longrightarrow> (Stars (v # vs)) \<succ>(STAR r) (Stars [])"
|
|
|
1877 |
| "\<lbrakk>v1 \<succ>r v2; v1 \<noteq> v2\<rbrakk> \<Longrightarrow> (Stars (v1 # vs1)) \<succ>(STAR r) (Stars (v2 # vs2))"
|
|
|
1878 |
| "(Stars vs1) \<succ>(STAR r) (Stars vs2) \<Longrightarrow> (Stars (v # vs1)) \<succ>(STAR r) (Stars (v # vs2))"
|
|
|
1879 |
| "(Stars []) \<succ>(STAR r) (Stars [])"
|
|
|
1880 |
|
|
|
1881 |
lemma PMatch_ValOrd:
|
|
|
1882 |
assumes "s \<in> r \<rightarrow> v" "v' \<in> SValues r s"
|
|
|
1883 |
shows "v \<succ>r v'"
|
|
|
1884 |
using assms
|
|
|
1885 |
apply(induct r arbitrary: v v' s rule: rexp.induct)
|
|
|
1886 |
apply(simp add: SValues_recs)
|
|
|
1887 |
apply(simp add: SValues_recs)
|
|
|
1888 |
apply(erule PMatch.cases)
|
|
|
1889 |
apply(simp_all)[7]
|
|
|
1890 |
apply (metis ValOrd.intros(7))
|
|
|
1891 |
apply(simp add: SValues_recs)
|
|
|
1892 |
apply(erule PMatch.cases)
|
|
|
1893 |
apply(simp_all)[7]
|
|
|
1894 |
apply (metis ValOrd.intros(8) empty_iff singletonD)
|
|
|
1895 |
apply(simp add: SValues_recs)
|
|
|
1896 |
apply(clarify)
|
|
|
1897 |
apply(erule PMatch.cases)
|
|
|
1898 |
apply(simp_all)[7]
|
|
|
1899 |
apply(clarify)
|
|
|
1900 |
apply(case_tac "v1a = v1")
|
|
|
1901 |
apply(simp)
|
|
|
1902 |
apply(rule ValOrd.intros)
|
|
|
1903 |
apply(rotate_tac 1)
|
|
|
1904 |
apply(drule_tac x="v2a" in meta_spec)
|
|
|
1905 |
apply(rotate_tac 8)
|
|
|
1906 |
apply(drule_tac x="v2" in meta_spec)
|
|
|
1907 |
apply(drule_tac x="s2a" in meta_spec)
|
|
|
1908 |
apply(simp)
|
|
|
1909 |
apply(drule_tac meta_mp)
|
|
|
1910 |
apply(simp add: SValues_def)
|
|
|
1911 |
apply (metis PMatch1(2) same_append_eq)
|
|
|
1912 |
apply(simp)
|
|
|
1913 |
apply(rule ValOrd.intros)
|
|
|
1914 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
1915 |
apply(rotate_tac 8)
|
|
|
1916 |
apply(drule_tac x="v1" in meta_spec)
|
|
|
1917 |
apply(drule_tac x="s1a" in meta_spec)
|
|
|
1918 |
apply(simp)
|
|
|
1919 |
apply(drule_tac meta_mp)
|
|
|
1920 |
apply(simp add: append_eq_append_conv2)
|
|
|
1921 |
apply(auto)[1]
|
|
|
1922 |
apply(case_tac "us=[]")
|
|
|
1923 |
apply(simp)
|
|
|
1924 |
apply(drule_tac x="us" in spec)
|
|
|
1925 |
apply(drule mp)
|
|
|
1926 |
apply(simp add: SValues_def)
|
|
|
1927 |
apply (metis Prf_flat_L)
|
|
|
1928 |
apply(erule disjE)
|
|
|
1929 |
apply(simp)
|
|
|
1930 |
apply(simp)
|
|
|
1931 |
apply(simp add: SValues_def)
|
|
|
1932 |
apply (metis Prf_flat_L)
|
|
|
1933 |
|
|
|
1934 |
apply(subst (asm) (2) Values_def)
|
|
|
1935 |
apply(simp)
|
|
|
1936 |
apply(clarify)
|
|
|
1937 |
apply(simp add: rest_def)
|
|
|
1938 |
apply(simp add: prefix_def)
|
|
|
1939 |
apply(auto)[1]
|
|
|
1940 |
apply(simp add: append_eq_append_conv2)
|
|
|
1941 |
apply(auto)[1]
|
|
|
1942 |
apply(case_tac "us = []")
|
|
|
1943 |
apply(simp)
|
|
|
1944 |
apply(simp add: Values_def)
|
|
|
1945 |
apply (metis append_Nil2 prefix_def)
|
|
|
1946 |
apply(drule_tac x="us" in spec)
|
|
|
1947 |
apply(simp)
|
|
|
1948 |
apply(drule_tac mp)
|
|
|
1949 |
|
|
|
1950 |
|
|
|
1951 |
oops
|
|
|
1952 |
(*HERE *)
|
|
|
1953 |
|
|
|
1954 |
inductive ValOrd2 :: "val \<Rightarrow> string \<Rightarrow> val \<Rightarrow> bool" ("_ 2\<succ>_ _" [100, 100, 100] 100)
|
|
|
1955 |
where
|
|
|
1956 |
"v2 2\<succ>s v2' \<Longrightarrow> (Seq v1 v2) 2\<succ>(flat v1 @ s) (Seq v1 v2')"
|
|
|
1957 |
| "\<lbrakk>v1 2\<succ>s v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) 2\<succ>s (Seq v1' v2')"
|
|
|
1958 |
| "(flat v2) \<sqsubseteq> (flat v1) \<Longrightarrow> (Left v1) 2\<succ>(flat v1) (Right v2)"
|
|
|
1959 |
| "(flat v1) \<sqsubset> (flat v2) \<Longrightarrow> (Right v2) 2\<succ>(flat v2) (Left v1)"
|
|
|
1960 |
| "v2 2\<succ>s v2' \<Longrightarrow> (Right v2) 2\<succ>s (Right v2')"
|
|
|
1961 |
| "v1 2\<succ>s v1' \<Longrightarrow> (Left v1) 2\<succ>s (Left v1')"
|
|
|
1962 |
| "Void 2\<succ>[] Void"
|
|
|
1963 |
| "(Char c) 2\<succ>[c] (Char c)"
|
|
|
1964 |
| "flat (Stars (v # vs)) = [] \<Longrightarrow> (Stars []) 2\<succ>[] (Stars (v # vs))"
|
|
|
1965 |
| "flat (Stars (v # vs)) \<noteq> [] \<Longrightarrow> (Stars (v # vs)) 2\<succ>(flat (Stars (v # vs))) (Stars [])"
|
|
|
1966 |
| "\<lbrakk>v1 2\<succ>s v2; v1 \<noteq> v2\<rbrakk> \<Longrightarrow> (Stars (v1 # vs1)) 2\<succ>s (Stars (v2 # vs2))"
|
|
|
1967 |
| "(Stars vs1) 2\<succ>s (Stars vs2) \<Longrightarrow> (Stars (v # vs1)) 2\<succ>(flat v @ s) (Stars (v # vs2))"
|
|
|
1968 |
| "(Stars []) 2\<succ>[] (Stars [])"
|
|
|
1969 |
|
|
|
1970 |
lemma ValOrd2_string1:
|
|
|
1971 |
assumes "v1 2\<succ>s v2"
|
|
|
1972 |
shows "s \<sqsubseteq> flat v1"
|
|
|
1973 |
using assms
|
|
|
1974 |
apply(induct)
|
|
|
1975 |
apply(auto simp add: prefix_def)
|
|
|
1976 |
apply (metis append_assoc)
|
|
|
1977 |
by (metis append_assoc)
|
|
|
1978 |
|
|
|
1979 |
|
|
|
1980 |
lemma admissibility:
|
|
|
1981 |
assumes "s \<in> r \<rightarrow> v" "\<turnstile> v' : r"
|
|
|
1982 |
shows "(\<forall>s'. (s' \<in> L(r) \<and> s' \<sqsubseteq> s) \<longrightarrow> v 2\<succ>s' v')"
|
|
|
1983 |
using assms
|
|
|
1984 |
apply(induct arbitrary: v')
|
|
|
1985 |
apply(erule Prf.cases)
|
|
|
1986 |
apply(simp_all)[7]
|
|
|
1987 |
apply (metis ValOrd2.intros(7))
|
|
|
1988 |
apply(erule Prf.cases)
|
|
|
1989 |
apply(simp_all)[7]
|
|
|
1990 |
apply (metis ValOrd2.intros(8) append_Nil2 prefix_Cons prefix_append prefix_def)
|
|
|
1991 |
apply(erule Prf.cases)
|
|
|
1992 |
apply(simp_all)[7]
|
|
|
1993 |
apply(auto)[1]
|
|
|
1994 |
apply (metis ValOrd2.intros(6))
|
|
|
1995 |
apply(rule ValOrd2.intros)
|
|
|
1996 |
apply(drule_tac x="v1" in meta_spec)
|
|
|
1997 |
apply(simp)
|
|
|
1998 |
|
|
|
1999 |
apply(clarify)
|
|
|
2000 |
apply (metis PMatch1(2) ValOrd2.intros(3))
|
|
|
2001 |
apply(erule Prf.cases)
|
|
|
2002 |
apply(simp_all)[7]
|
|
|
2003 |
apply(auto)
|
|
|
2004 |
|
|
|
2005 |
apply(case_tac "v1 = v1a")
|
|
|
2006 |
apply(simp)
|
|
|
2007 |
apply(rotate_tac 3)
|
|
|
2008 |
apply(drule_tac x="v2a" in meta_spec)
|
|
|
2009 |
apply(drule meta_mp)
|
|
|
2010 |
apply(simp)
|
|
|
2011 |
apply(auto)
|
|
|
2012 |
apply(rule_tac x="flat v1a @ s'" in exI)
|
|
|
2013 |
apply (metis PMatch1(2) ValOrd2.intros(1) prefix_append)
|
|
|
2014 |
apply (metis PMatch1(2) ValOrd2.intros(2) ValOrd2_string1 flat.simps(5))
|
|
|
2015 |
prefer 4
|
|
|
2016 |
apply(erule Prf.cases)
|
|
|
2017 |
apply(simp_all)[7]
|
|
|
2018 |
prefer 2
|
|
|
2019 |
apply (metis ValOrd2.intros(5))
|
|
|
2020 |
|
|
|
2021 |
|
|
|
2022 |
apply (metis ValOrd.intros(6))
|
|
|
2023 |
oops
|
|
|
2024 |
|
|
|
2025 |
|
|
|
2026 |
lemma admissibility:
|
|
|
2027 |
assumes "\<turnstile> s \<in> r \<rightarrow> v" "\<turnstile> v' : r"
|
|
|
2028 |
shows "v \<succ>r v'"
|
|
|
2029 |
using assms
|
|
|
2030 |
apply(induct arbitrary: v')
|
|
|
2031 |
apply(erule Prf.cases)
|
|
|
2032 |
apply(simp_all)[7]
|
|
|
2033 |
apply (metis ValOrd.intros(7))
|
|
|
2034 |
apply(erule Prf.cases)
|
|
|
2035 |
apply(simp_all)[7]
|
|
|
2036 |
apply (metis ValOrd.intros(8))
|
|
|
2037 |
apply(erule Prf.cases)
|
|
|
2038 |
apply(simp_all)[7]
|
|
|
2039 |
apply (metis ValOrd.intros(6))
|
|
|
2040 |
oops
|
|
|
2041 |
|
|
|
2042 |
lemma admissibility:
|
|
|
2043 |
assumes "2\<rhd> v : r" "\<turnstile> v' : r" "flat v' \<sqsubseteq> flat v"
|
|
|
2044 |
shows "v \<succ>r v'"
|
|
|
2045 |
using assms
|
|
|
2046 |
apply(induct arbitrary: v')
|
|
|
2047 |
apply(erule Prf.cases)
|
|
|
2048 |
apply(simp_all)[7]
|
|
|
2049 |
apply (metis ValOrd.intros(7))
|
|
|
2050 |
apply(erule Prf.cases)
|
|
|
2051 |
apply(simp_all)[7]
|
|
|
2052 |
apply (metis ValOrd.intros(8))
|
|
|
2053 |
apply(erule Prf.cases)
|
|
|
2054 |
apply(simp_all)[7]
|
|
|
2055 |
apply (metis ValOrd.intros(6))
|
|
|
2056 |
apply (metis ValOrd.intros(3) length_sprefix less_imp_le_nat order_refl sprefix_def)
|
|
|
2057 |
apply(erule Prf.cases)
|
|
|
2058 |
apply(simp_all)[7]
|
|
|
2059 |
apply (metis Prf_flat_L ValOrd.intros(4) length_sprefix seq_empty(1) sprefix_def)
|
|
|
2060 |
apply (metis ValOrd.intros(5))
|
|
|
2061 |
oops
|
|
|
2062 |
|
|
|
2063 |
|
|
|
2064 |
lemma admisibility:
|
|
|
2065 |
assumes "\<rhd> v : r" "\<turnstile> v' : r"
|
|
|
2066 |
shows "v \<succ>r v'"
|
|
|
2067 |
using assms
|
|
|
2068 |
apply(induct arbitrary: v')
|
|
|
2069 |
prefer 5
|
|
|
2070 |
apply(drule royA)
|
|
|
2071 |
apply(erule Prf.cases)
|
|
|
2072 |
apply(simp_all)[7]
|
|
|
2073 |
apply(clarify)
|
|
|
2074 |
apply(case_tac "v1 = v1a")
|
|
|
2075 |
apply(simp)
|
|
|
2076 |
apply(rule ValOrd.intros)
|
|
|
2077 |
apply metis
|
|
|
2078 |
apply (metis ValOrd.intros(2))
|
|
|
2079 |
apply(erule Prf.cases)
|
|
|
2080 |
apply(simp_all)[7]
|
|
|
2081 |
apply (metis ValOrd.intros(7))
|
|
|
2082 |
apply(erule Prf.cases)
|
|
|
2083 |
apply(simp_all)[7]
|
|
|
2084 |
apply (metis ValOrd.intros(8))
|
|
|
2085 |
apply(erule Prf.cases)
|
|
|
2086 |
apply(simp_all)[7]
|
|
|
2087 |
apply (metis ValOrd.intros(6))
|
|
|
2088 |
apply(rule ValOrd.intros)
|
|
|
2089 |
defer
|
|
|
2090 |
apply(erule Prf.cases)
|
|
|
2091 |
apply(simp_all)[7]
|
|
|
2092 |
apply(clarify)
|
|
|
2093 |
apply(rule ValOrd.intros)
|
|
|
2094 |
(* seq case goes through *)
|
|
|
2095 |
oops
|
|
|
2096 |
|
|
|
2097 |
|
|
|
2098 |
lemma admisibility:
|
|
|
2099 |
assumes "\<rhd> v : r" "\<turnstile> v' : r" "flat v' \<sqsubseteq> flat v"
|
|
|
2100 |
shows "v \<succ>r v'"
|
|
|
2101 |
using assms
|
|
|
2102 |
apply(induct arbitrary: v')
|
|
|
2103 |
prefer 5
|
|
|
2104 |
apply(drule royA)
|
|
|
2105 |
apply(erule Prf.cases)
|
|
|
2106 |
apply(simp_all)[7]
|
|
|
2107 |
apply(clarify)
|
|
|
2108 |
apply(case_tac "v1 = v1a")
|
|
|
2109 |
apply(simp)
|
|
|
2110 |
apply(rule ValOrd.intros)
|
|
|
2111 |
apply(subst (asm) (3) prefix_def)
|
|
|
2112 |
apply(erule exE)
|
|
|
2113 |
apply(simp)
|
|
|
2114 |
apply (metis prefix_def)
|
|
|
2115 |
(* the unequal case *)
|
|
|
2116 |
apply(subgoal_tac "flat v1 \<sqsubset> flat v1a \<or> flat v1a \<sqsubseteq> flat v1")
|
|
|
2117 |
prefer 2
|
|
|
2118 |
apply(simp add: prefix_def sprefix_def)
|
|
|
2119 |
apply (metis append_eq_append_conv2)
|
|
|
2120 |
apply(erule disjE)
|
|
|
2121 |
(* first case flat v1 \<sqsubset> flat v1a *)
|
|
|
2122 |
apply(subst (asm) sprefix_def)
|
|
|
2123 |
apply(subst (asm) (5) prefix_def)
|
|
|
2124 |
apply(clarify)
|
|
|
2125 |
apply(subgoal_tac "(s3 @ flat v2a) \<sqsubseteq> flat v2")
|
|
|
2126 |
prefer 2
|
|
|
2127 |
apply(simp)
|
|
|
2128 |
apply (metis append_assoc prefix_append)
|
|
|
2129 |
apply(subgoal_tac "s3 \<noteq> []")
|
|
|
2130 |
prefer 2
|
|
|
2131 |
apply (metis append_Nil2)
|
|
|
2132 |
(* HERE *)
|
|
|
2133 |
apply(subst (asm) (5) prefix_def)
|
|
|
2134 |
apply(erule exE)
|
|
|
2135 |
apply(simp add: ders_correctness Ders_def)
|
|
|
2136 |
apply(simp add: prefix_def)
|
|
|
2137 |
apply(clarify)
|
|
|
2138 |
apply(subst (asm) append_eq_append_conv2)
|
|
|
2139 |
apply(erule exE)
|
|
|
2140 |
apply(erule disjE)
|
|
|
2141 |
apply(clarify)
|
|
|
2142 |
oops
|
|
|
2143 |
|
|
|
2144 |
|
|
|
2145 |
|
|
|
2146 |
lemma ValOrd_refl:
|
|
|
2147 |
assumes "\<turnstile> v : r"
|
|
|
2148 |
shows "v \<succ>r v"
|
|
|
2149 |
using assms
|
|
|
2150 |
apply(induct)
|
|
|
2151 |
apply(auto intro: ValOrd.intros)
|
|
|
2152 |
done
|
|
|
2153 |
|
|
|
2154 |
lemma ValOrd_total:
|
|
|
2155 |
shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r\<rbrakk> \<Longrightarrow> v1 \<succ>r v2 \<or> v2 \<succ>r v1"
|
|
|
2156 |
apply(induct r arbitrary: v1 v2)
|
|
|
2157 |
apply(auto)
|
|
|
2158 |
apply(erule Prf.cases)
|
|
|
2159 |
apply(simp_all)[7]
|
|
|
2160 |
apply(erule Prf.cases)
|
|
|
2161 |
apply(simp_all)[7]
|
|
|
2162 |
apply(erule Prf.cases)
|
|
|
2163 |
apply(simp_all)[7]
|
|
|
2164 |
apply (metis ValOrd.intros(7))
|
|
|
2165 |
apply(erule Prf.cases)
|
|
|
2166 |
apply(simp_all)[7]
|
|
|
2167 |
apply(erule Prf.cases)
|
|
|
2168 |
apply(simp_all)[7]
|
|
|
2169 |
apply (metis ValOrd.intros(8))
|
|
|
2170 |
apply(erule Prf.cases)
|
|
|
2171 |
apply(simp_all)[7]
|
|
|
2172 |
apply(erule Prf.cases)
|
|
|
2173 |
apply(simp_all)[7]
|
|
|
2174 |
apply(clarify)
|
|
|
2175 |
apply(case_tac "v1a = v1b")
|
|
|
2176 |
apply(simp)
|
|
|
2177 |
apply(rule ValOrd.intros(1))
|
|
|
2178 |
apply (metis ValOrd.intros(1))
|
|
|
2179 |
apply(rule ValOrd.intros(2))
|
|
|
2180 |
apply(auto)[2]
|
|
|
2181 |
apply(erule contrapos_np)
|
|
|
2182 |
apply(rule ValOrd.intros(2))
|
|
|
2183 |
apply(auto)
|
|
|
2184 |
apply(erule Prf.cases)
|
|
|
2185 |
apply(simp_all)[7]
|
|
|
2186 |
apply(erule Prf.cases)
|
|
|
2187 |
apply(simp_all)[7]
|
|
|
2188 |
apply(clarify)
|
|
|
2189 |
apply (metis ValOrd.intros(6))
|
|
|
2190 |
apply(rule ValOrd.intros)
|
|
|
2191 |
apply(erule contrapos_np)
|
|
|
2192 |
apply(rule ValOrd.intros)
|
|
|
2193 |
apply (metis le_eq_less_or_eq neq_iff)
|
|
|
2194 |
apply(erule Prf.cases)
|
|
|
2195 |
apply(simp_all)[7]
|
|
|
2196 |
apply(rule ValOrd.intros)
|
|
|
2197 |
apply(erule contrapos_np)
|
|
|
2198 |
apply(rule ValOrd.intros)
|
|
|
2199 |
apply (metis le_eq_less_or_eq neq_iff)
|
|
|
2200 |
apply(rule ValOrd.intros)
|
|
|
2201 |
apply(erule contrapos_np)
|
|
|
2202 |
apply(rule ValOrd.intros)
|
|
|
2203 |
apply(metis)
|
|
|
2204 |
apply(erule Prf.cases)
|
|
|
2205 |
apply(simp_all)[7]
|
|
|
2206 |
apply(erule Prf.cases)
|
|
|
2207 |
apply(simp_all)[7]
|
|
|
2208 |
apply(auto)
|
|
|
2209 |
apply (metis ValOrd.intros(13))
|
|
|
2210 |
apply (metis ValOrd.intros(10) ValOrd.intros(9))
|
|
|
2211 |
apply(erule Prf.cases)
|
|
|
2212 |
apply(simp_all)[7]
|
|
|
2213 |
apply(auto)
|
|
|
2214 |
apply (metis ValOrd.intros(10) ValOrd.intros(9))
|
|
|
2215 |
apply(case_tac "v = va")
|
|
|
2216 |
prefer 2
|
|
|
2217 |
apply (metis ValOrd.intros(11))
|
|
|
2218 |
apply(simp)
|
|
|
2219 |
apply(rule ValOrd.intros(12))
|
|
|
2220 |
apply(erule contrapos_np)
|
|
|
2221 |
apply(rule ValOrd.intros(12))
|
|
|
2222 |
oops
|
|
|
2223 |
|
|
|
2224 |
lemma Roy_posix:
|
|
|
2225 |
assumes "\<rhd> v : r" "\<turnstile> v' : r" "flat v' \<sqsubseteq> flat v"
|
|
|
2226 |
shows "v \<succ>r v'"
|
|
|
2227 |
using assms
|
|
|
2228 |
apply(induct r arbitrary: v v' rule: rexp.induct)
|
|
|
2229 |
apply(erule Prf.cases)
|
|
|
2230 |
apply(simp_all)[7]
|
|
|
2231 |
apply(erule Prf.cases)
|
|
|
2232 |
apply(simp_all)[7]
|
|
|
2233 |
apply(erule Roy.cases)
|
|
|
2234 |
apply(simp_all)
|
|
|
2235 |
apply (metis ValOrd.intros(7))
|
|
|
2236 |
apply(erule Prf.cases)
|
|
|
2237 |
apply(simp_all)[7]
|
|
|
2238 |
apply(erule Roy.cases)
|
|
|
2239 |
apply(simp_all)
|
|
|
2240 |
apply (metis ValOrd.intros(8))
|
|
|
2241 |
prefer 2
|
|
|
2242 |
apply(erule Prf.cases)
|
|
|
2243 |
apply(simp_all)[7]
|
|
|
2244 |
apply(erule Roy.cases)
|
|
|
2245 |
apply(simp_all)
|
|
|
2246 |
apply(clarify)
|
|
|
2247 |
apply (metis ValOrd.intros(6))
|
|
|
2248 |
apply(clarify)
|
|
|
2249 |
apply (metis Prf_flat_L ValOrd.intros(4) length_sprefix sprefix_def)
|
|
|
2250 |
apply(erule Roy.cases)
|
|
|
2251 |
apply(simp_all)
|
|
|
2252 |
apply (metis ValOrd.intros(3) length_sprefix less_imp_le_nat order_refl sprefix_def)
|
|
|
2253 |
apply(clarify)
|
|
|
2254 |
apply (metis ValOrd.intros(5))
|
|
|
2255 |
apply(erule Prf.cases)
|
|
|
2256 |
apply(simp_all)[7]
|
|
|
2257 |
apply(erule Roy.cases)
|
|
|
2258 |
apply(simp_all)
|
|
|
2259 |
apply(clarify)
|
|
|
2260 |
apply(case_tac "v1a = v1")
|
|
|
2261 |
apply(simp)
|
|
|
2262 |
apply(rule ValOrd.intros)
|
|
|
2263 |
apply (metis prefix_append)
|
|
|
2264 |
apply(rule ValOrd.intros)
|
|
|
2265 |
prefer 2
|
|
|
2266 |
apply(simp)
|
|
|
2267 |
apply(simp add: prefix_def)
|
|
|
2268 |
apply(auto)[1]
|
|
|
2269 |
apply(simp add: append_eq_append_conv2)
|
|
|
2270 |
apply(auto)[1]
|
|
|
2271 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
2272 |
apply(rotate_tac 9)
|
|
|
2273 |
apply(drule_tac x="v1" in meta_spec)
|
|
|
2274 |
apply(drule_tac meta_mp)
|
|
|
2275 |
apply(simp)
|
|
|
2276 |
apply(drule_tac meta_mp)
|
|
|
2277 |
apply(simp)
|
|
|
2278 |
apply(drule_tac meta_mp)
|
|
|
2279 |
apply(simp)
|
|
|
2280 |
apply(drule_tac x="us" in spec)
|
|
|
2281 |
apply(drule_tac mp)
|
|
|
2282 |
apply (metis Prf_flat_L)
|
|
|
2283 |
apply(auto)[1]
|
|
|
2284 |
oops
|
|
|
2285 |
|
|
|
2286 |
|
|
|
2287 |
lemma ValOrd_anti:
|
|
|
2288 |
shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r; v1 \<succ>r v2; v2 \<succ>r v1\<rbrakk> \<Longrightarrow> v1 = v2"
|
|
|
2289 |
and "\<lbrakk>\<turnstile> Stars vs1 : r; \<turnstile> Stars vs2 : r; Stars vs1 \<succ>r Stars vs2; Stars vs2 \<succ>r Stars vs1\<rbrakk> \<Longrightarrow> vs1 = vs2"
|
|
|
2290 |
apply(induct v1 and vs1 arbitrary: r v2 and r vs2 rule: val.inducts)
|
|
|
2291 |
apply(erule Prf.cases)
|
|
|
2292 |
apply(simp_all)[7]
|
|
|
2293 |
apply(erule Prf.cases)
|
|
|
2294 |
apply(simp_all)[7]
|
|
|
2295 |
apply(erule Prf.cases)
|
|
|
2296 |
apply(simp_all)[7]
|
|
|
2297 |
apply(erule Prf.cases)
|
|
|
2298 |
apply(simp_all)[7]
|
|
|
2299 |
apply(erule Prf.cases)
|
|
|
2300 |
apply(simp_all)[7]
|
|
|
2301 |
apply(erule Prf.cases)
|
|
|
2302 |
apply(simp_all)[7]
|
|
|
2303 |
apply(erule ValOrd.cases)
|
|
|
2304 |
apply(simp_all)
|
|
|
2305 |
apply(erule ValOrd.cases)
|
|
|
2306 |
apply(simp_all)
|
|
|
2307 |
apply(erule ValOrd.cases)
|
|
|
2308 |
apply(simp_all)
|
|
|
2309 |
apply(erule Prf.cases)
|
|
|
2310 |
apply(simp_all)[7]
|
|
|
2311 |
apply(erule Prf.cases)
|
|
|
2312 |
apply(simp_all)[7]
|
|
|
2313 |
apply(erule ValOrd.cases)
|
|
|
2314 |
apply(simp_all)
|
|
|
2315 |
apply(erule ValOrd.cases)
|
|
|
2316 |
apply(simp_all)
|
|
|
2317 |
apply(erule ValOrd.cases)
|
|
|
2318 |
apply(simp_all)
|
|
|
2319 |
apply(erule ValOrd.cases)
|
|
|
2320 |
apply(simp_all)
|
|
|
2321 |
apply(erule Prf.cases)
|
|
|
2322 |
apply(simp_all)[7]
|
|
|
2323 |
apply(erule Prf.cases)
|
|
|
2324 |
apply(simp_all)[7]
|
|
|
2325 |
apply(erule ValOrd.cases)
|
|
|
2326 |
apply(simp_all)
|
|
|
2327 |
apply(erule ValOrd.cases)
|
|
|
2328 |
apply(simp_all)
|
|
|
2329 |
apply(erule ValOrd.cases)
|
|
|
2330 |
apply(simp_all)
|
|
|
2331 |
apply(erule ValOrd.cases)
|
|
|
2332 |
apply(simp_all)
|
|
|
2333 |
apply(erule Prf.cases)
|
|
|
2334 |
apply(simp_all)[7]
|
|
|
2335 |
apply(erule Prf.cases)
|
|
|
2336 |
apply(simp_all)[7]
|
|
|
2337 |
apply(erule ValOrd.cases)
|
|
|
2338 |
apply(simp_all)
|
|
|
2339 |
apply(erule ValOrd.cases)
|
|
|
2340 |
apply(simp_all)
|
|
|
2341 |
apply(erule Prf.cases)
|
|
|
2342 |
apply(simp_all)[7]
|
|
|
2343 |
apply(erule ValOrd.cases)
|
|
|
2344 |
apply(simp_all)
|
|
|
2345 |
apply(erule ValOrd.cases)
|
|
|
2346 |
apply(simp_all)
|
|
|
2347 |
apply(erule ValOrd.cases)
|
|
|
2348 |
apply(simp_all)
|
|
|
2349 |
apply(erule ValOrd.cases)
|
|
|
2350 |
apply(simp_all)
|
|
|
2351 |
apply(auto)[1]
|
|
|
2352 |
prefer 2
|
|
|
2353 |
oops
|
|
|
2354 |
|
|
|
2355 |
|
|
|
2356 |
(*
|
|
|
2357 |
|
|
|
2358 |
lemma ValOrd_PMatch:
|
|
|
2359 |
assumes "s \<in> r \<rightarrow> v1" "\<turnstile> v2 : r" "flat v2 \<sqsubseteq> s"
|
|
|
2360 |
shows "v1 \<succ>r v2"
|
|
|
2361 |
using assms
|
|
|
2362 |
apply(induct r arbitrary: s v1 v2 rule: rexp.induct)
|
|
|
2363 |
apply(erule Prf.cases)
|
|
|
2364 |
apply(simp_all)[7]
|
|
|
2365 |
apply(erule Prf.cases)
|
|
|
2366 |
apply(simp_all)[7]
|
|
|
2367 |
apply(erule PMatch.cases)
|
|
|
2368 |
apply(simp_all)[7]
|
|
|
2369 |
apply (metis ValOrd.intros(7))
|
|
|
2370 |
apply(erule Prf.cases)
|
|
|
2371 |
apply(simp_all)[7]
|
|
|
2372 |
apply(erule PMatch.cases)
|
|
|
2373 |
apply(simp_all)[7]
|
|
|
2374 |
apply (metis ValOrd.intros(8))
|
|
|
2375 |
defer
|
|
|
2376 |
apply(erule Prf.cases)
|
|
|
2377 |
apply(simp_all)[7]
|
|
|
2378 |
apply(erule PMatch.cases)
|
|
|
2379 |
apply(simp_all)[7]
|
|
|
2380 |
apply (metis ValOrd.intros(6))
|
|
|
2381 |
apply (metis PMatch1(2) Prf_flat_L ValOrd.intros(4) length_sprefix sprefix_def)
|
|
|
2382 |
apply(clarify)
|
|
|
2383 |
apply(erule PMatch.cases)
|
|
|
2384 |
apply(simp_all)[7]
|
|
|
2385 |
apply (metis PMatch1(2) ValOrd.intros(3) length_sprefix less_imp_le_nat order_refl sprefix_def)
|
|
|
2386 |
apply(clarify)
|
|
|
2387 |
apply (metis ValOrd.intros(5))
|
|
|
2388 |
(* Stars case *)
|
|
|
2389 |
apply(erule Prf.cases)
|
|
|
2390 |
apply(simp_all)[7]
|
|
|
2391 |
apply(erule PMatch.cases)
|
|
|
2392 |
apply(simp_all)
|
|
|
2393 |
apply (metis Nil_is_append_conv ValOrd.intros(10) flat.simps(7))
|
|
|
2394 |
apply (metis ValOrd.intros(13))
|
|
|
2395 |
apply(clarify)
|
|
|
2396 |
apply(erule PMatch.cases)
|
|
|
2397 |
apply(simp_all)
|
|
|
2398 |
prefer 2
|
|
|
2399 |
apply(rule ValOrd.intros)
|
|
|
2400 |
apply(simp add: prefix_def)
|
|
|
2401 |
apply(rule ValOrd.intros)
|
|
|
2402 |
apply(drule_tac x="s1" in meta_spec)
|
|
|
2403 |
apply(drule_tac x="va" in meta_spec)
|
|
|
2404 |
apply(drule_tac x="v" in meta_spec)
|
|
|
2405 |
apply(drule_tac meta_mp)
|
|
|
2406 |
apply(simp)
|
|
|
2407 |
apply(drule_tac meta_mp)
|
|
|
2408 |
apply(simp)
|
|
|
2409 |
apply(drule_tac meta_mp)
|
|
|
2410 |
apply(simp add: prefix_def)
|
|
|
2411 |
apply(auto)[1]
|
|
|
2412 |
prefer 3
|
|
|
2413 |
(* Seq case *)
|
|
|
2414 |
apply(erule Prf.cases)
|
|
|
2415 |
apply(simp_all)[5]
|
|
|
2416 |
apply(auto)
|
|
|
2417 |
apply(erule PMatch.cases)
|
|
|
2418 |
apply(simp_all)[5]
|
|
|
2419 |
apply(auto)
|
|
|
2420 |
apply(case_tac "v1b = v1a")
|
|
|
2421 |
apply(auto)
|
|
|
2422 |
apply(simp add: prefix_def)
|
|
|
2423 |
apply(auto)[1]
|
|
|
2424 |
apply (metis PMatch1(2) ValOrd.intros(1) same_append_eq)
|
|
|
2425 |
apply(simp add: prefix_def)
|
|
|
2426 |
apply(auto)[1]
|
|
|
2427 |
apply(simp add: append_eq_append_conv2)
|
|
|
2428 |
apply(auto)
|
|
|
2429 |
prefer 2
|
|
|
2430 |
apply (metis ValOrd.intros(2))
|
|
|
2431 |
prefer 2
|
|
|
2432 |
apply (metis ValOrd.intros(2))
|
|
|
2433 |
apply(case_tac "us = []")
|
|
|
2434 |
apply(simp)
|
|
|
2435 |
apply (metis ValOrd.intros(2) append_Nil2)
|
|
|
2436 |
apply(drule_tac x="us" in spec)
|
|
|
2437 |
apply(simp)
|
|
|
2438 |
apply(drule_tac mp)
|
|
|
2439 |
apply (metis Prf_flat_L)
|
|
|
2440 |
apply(drule_tac x="s1 @ us" in meta_spec)
|
|
|
2441 |
apply(drule_tac x="v1b" in meta_spec)
|
|
|
2442 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
2443 |
apply(drule_tac meta_mp)
|
|
|
2444 |
|
|
|
2445 |
apply(simp)
|
|
|
2446 |
apply(drule_tac meta_mp)
|
|
|
2447 |
apply(simp)
|
|
|
2448 |
apply(simp)
|
|
|
2449 |
apply(simp)
|
|
|
2450 |
apply(clarify)
|
|
|
2451 |
apply (metis ValOrd.intros(6))
|
|
|
2452 |
apply(clarify)
|
|
|
2453 |
apply (metis PMatch1(2) ValOrd.intros(3) length_sprefix less_imp_le_nat order_refl sprefix_def)
|
|
|
2454 |
apply(erule Prf.cases)
|
|
|
2455 |
apply(simp_all)[5]
|
|
|
2456 |
apply(clarify)
|
|
|
2457 |
apply (metis PMatch1(2) Prf_flat_L ValOrd.intros(4) length_sprefix sprefix_def)
|
|
|
2458 |
apply (metis ValOrd.intros(5))
|
|
|
2459 |
(* Seq case *)
|
|
|
2460 |
apply(erule Prf.cases)
|
|
|
2461 |
apply(simp_all)[5]
|
|
|
2462 |
apply(auto)
|
|
|
2463 |
apply(case_tac "v1 = v1a")
|
|
|
2464 |
apply(auto)
|
|
|
2465 |
apply(simp add: prefix_def)
|
|
|
2466 |
apply(auto)[1]
|
|
|
2467 |
apply (metis PMatch1(2) ValOrd.intros(1) same_append_eq)
|
|
|
2468 |
apply(simp add: prefix_def)
|
|
|
2469 |
apply(auto)[1]
|
|
|
2470 |
apply(frule PMatch1)
|
|
|
2471 |
apply(frule PMatch1(2)[symmetric])
|
|
|
2472 |
apply(clarify)
|
|
|
2473 |
apply(simp add: append_eq_append_conv2)
|
|
|
2474 |
apply(auto)
|
|
|
2475 |
prefer 2
|
|
|
2476 |
apply (metis ValOrd.intros(2))
|
|
|
2477 |
prefer 2
|
|
|
2478 |
apply (metis ValOrd.intros(2))
|
|
|
2479 |
apply(case_tac "us = []")
|
|
|
2480 |
apply(simp)
|
|
|
2481 |
apply (metis ValOrd.intros(2) append_Nil2)
|
|
|
2482 |
apply(drule_tac x="us" in spec)
|
|
|
2483 |
apply(simp)
|
|
|
2484 |
apply(drule mp)
|
|
|
2485 |
apply (metis Prf_flat_L)
|
|
|
2486 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
2487 |
apply(drule_tac meta_mp)
|
|
|
2488 |
apply(simp)
|
|
|
2489 |
apply(drule_tac meta_mp)
|
|
|
2490 |
apply(simp)
|
|
|
2491 |
|
|
|
2492 |
lemma ValOrd_PMatch:
|
|
|
2493 |
assumes "s \<in> r \<rightarrow> v1" "\<turnstile> v2 : r" "flat v2 \<sqsubseteq> s"
|
|
|
2494 |
shows "v1 \<succ>r v2"
|
|
|
2495 |
using assms
|
|
|
2496 |
apply(induct arbitrary: v2 rule: .induct)
|
|
|
2497 |
apply(erule Prf.cases)
|
|
|
2498 |
apply(simp_all)[5]
|
|
|
2499 |
apply (metis ValOrd.intros(7))
|
|
|
2500 |
apply(erule Prf.cases)
|
|
|
2501 |
apply(simp_all)[5]
|
|
|
2502 |
apply (metis ValOrd.intros(8))
|
|
|
2503 |
apply(erule Prf.cases)
|
|
|
2504 |
apply(simp_all)[5]
|
|
|
2505 |
apply(clarify)
|
|
|
2506 |
apply (metis ValOrd.intros(6))
|
|
|
2507 |
apply(clarify)
|
|
|
2508 |
apply (metis PMatch1(2) ValOrd.intros(3) length_sprefix less_imp_le_nat order_refl sprefix_def)
|
|
|
2509 |
apply(erule Prf.cases)
|
|
|
2510 |
apply(simp_all)[5]
|
|
|
2511 |
apply(clarify)
|
|
|
2512 |
apply (metis PMatch1(2) Prf_flat_L ValOrd.intros(4) length_sprefix sprefix_def)
|
|
|
2513 |
apply (metis ValOrd.intros(5))
|
|
|
2514 |
(* Seq case *)
|
|
|
2515 |
apply(erule Prf.cases)
|
|
|
2516 |
apply(simp_all)[5]
|
|
|
2517 |
apply(auto)
|
|
|
2518 |
apply(case_tac "v1 = v1a")
|
|
|
2519 |
apply(auto)
|
|
|
2520 |
apply(simp add: prefix_def)
|
|
|
2521 |
apply(auto)[1]
|
|
|
2522 |
apply (metis PMatch1(2) ValOrd.intros(1) same_append_eq)
|
|
|
2523 |
apply(simp add: prefix_def)
|
|
|
2524 |
apply(auto)[1]
|
|
|
2525 |
apply(frule PMatch1)
|
|
|
2526 |
apply(frule PMatch1(2)[symmetric])
|
|
|
2527 |
apply(clarify)
|
|
|
2528 |
apply(simp add: append_eq_append_conv2)
|
|
|
2529 |
apply(auto)
|
|
|
2530 |
prefer 2
|
|
|
2531 |
apply (metis ValOrd.intros(2))
|
|
|
2532 |
prefer 2
|
|
|
2533 |
apply (metis ValOrd.intros(2))
|
|
|
2534 |
apply(case_tac "us = []")
|
|
|
2535 |
apply(simp)
|
|
|
2536 |
apply (metis ValOrd.intros(2) append_Nil2)
|
|
|
2537 |
apply(drule_tac x="us" in spec)
|
|
|
2538 |
apply(simp)
|
|
|
2539 |
apply(drule mp)
|
|
|
2540 |
apply (metis Prf_flat_L)
|
|
|
2541 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
2542 |
apply(drule_tac meta_mp)
|
|
|
2543 |
apply(simp)
|
|
|
2544 |
apply(drule_tac meta_mp)
|
|
|
2545 |
apply(simp)
|
|
|
2546 |
|
|
|
2547 |
apply (metis PMatch1(2) ValOrd.intros(1) same_append_eq)
|
|
|
2548 |
apply(rule ValOrd.intros(2))
|
|
|
2549 |
apply(auto)
|
|
|
2550 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
2551 |
apply(drule_tac meta_mp)
|
|
|
2552 |
apply(simp)
|
|
|
2553 |
apply(drule_tac meta_mp)
|
|
|
2554 |
prefer 2
|
|
|
2555 |
apply(simp)
|
|
|
2556 |
thm append_eq_append_conv
|
|
|
2557 |
apply(simp add: append_eq_append_conv2)
|
|
|
2558 |
apply(auto)
|
|
|
2559 |
apply (metis Prf_flat_L)
|
|
|
2560 |
apply(case_tac "us = []")
|
|
|
2561 |
apply(simp)
|
|
|
2562 |
apply(drule_tac x="us" in spec)
|
|
|
2563 |
apply(drule mp)
|
|
|
2564 |
|
|
|
2565 |
|
|
|
2566 |
inductive ValOrd2 :: "val \<Rightarrow> val \<Rightarrow> bool" ("_ 2\<succ> _" [100, 100] 100)
|
|
|
2567 |
where
|
|
|
2568 |
"v2 2\<succ> v2' \<Longrightarrow> (Seq v1 v2) 2\<succ> (Seq v1 v2')"
|
|
|
2569 |
| "\<lbrakk>v1 2\<succ> v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) 2\<succ> (Seq v1' v2')"
|
|
|
2570 |
| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) 2\<succ> (Right v2)"
|
|
|
2571 |
| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) 2\<succ> (Left v1)"
|
|
|
2572 |
| "v2 2\<succ> v2' \<Longrightarrow> (Right v2) 2\<succ> (Right v2')"
|
|
|
2573 |
| "v1 2\<succ> v1' \<Longrightarrow> (Left v1) 2\<succ> (Left v1')"
|
|
|
2574 |
| "Void 2\<succ> Void"
|
|
|
2575 |
| "(Char c) 2\<succ> (Char c)"
|
|
|
2576 |
|
|
|
2577 |
lemma Ord1:
|
|
|
2578 |
"v1 \<succ>r v2 \<Longrightarrow> v1 2\<succ> v2"
|
|
|
2579 |
apply(induct rule: ValOrd.induct)
|
|
|
2580 |
apply(auto intro: ValOrd2.intros)
|
|
|
2581 |
done
|
|
|
2582 |
|
|
|
2583 |
lemma Ord2:
|
|
|
2584 |
"v1 2\<succ> v2 \<Longrightarrow> \<exists>r. v1 \<succ>r v2"
|
|
|
2585 |
apply(induct v1 v2 rule: ValOrd2.induct)
|
|
|
2586 |
apply(auto intro: ValOrd.intros)
|
|
|
2587 |
done
|
|
|
2588 |
|
|
|
2589 |
lemma Ord3:
|
|
|
2590 |
"\<lbrakk>v1 2\<succ> v2; \<turnstile> v1 : r\<rbrakk> \<Longrightarrow> v1 \<succ>r v2"
|
|
|
2591 |
apply(induct v1 v2 arbitrary: r rule: ValOrd2.induct)
|
|
|
2592 |
apply(auto intro: ValOrd.intros elim: Prf.cases)
|
|
|
2593 |
done
|
|
|
2594 |
|
|
|
2595 |
section {* Posix definition *}
|
|
|
2596 |
|
|
|
2597 |
definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
2598 |
where
|
|
|
2599 |
"POSIX v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v' \<sqsubseteq> flat v) \<longrightarrow> v \<succ>r v'))"
|
|
|
2600 |
|
|
|
2601 |
lemma ValOrd_refl:
|
|
|
2602 |
assumes "\<turnstile> v : r"
|
|
|
2603 |
shows "v \<succ>r v"
|
|
|
2604 |
using assms
|
|
|
2605 |
apply(induct)
|
|
|
2606 |
apply(auto intro: ValOrd.intros)
|
|
|
2607 |
done
|
|
|
2608 |
|
|
|
2609 |
lemma ValOrd_total:
|
|
|
2610 |
shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r\<rbrakk> \<Longrightarrow> v1 \<succ>r v2 \<or> v2 \<succ>r v1"
|
|
|
2611 |
apply(induct r arbitrary: v1 v2)
|
|
|
2612 |
apply(auto)
|
|
|
2613 |
apply(erule Prf.cases)
|
|
|
2614 |
apply(simp_all)[5]
|
|
|
2615 |
apply(erule Prf.cases)
|
|
|
2616 |
apply(simp_all)[5]
|
|
|
2617 |
apply(erule Prf.cases)
|
|
|
2618 |
apply(simp_all)[5]
|
|
|
2619 |
apply (metis ValOrd.intros(7))
|
|
|
2620 |
apply(erule Prf.cases)
|
|
|
2621 |
apply(simp_all)[5]
|
|
|
2622 |
apply(erule Prf.cases)
|
|
|
2623 |
apply(simp_all)[5]
|
|
|
2624 |
apply (metis ValOrd.intros(8))
|
|
|
2625 |
apply(erule Prf.cases)
|
|
|
2626 |
apply(simp_all)[5]
|
|
|
2627 |
apply(erule Prf.cases)
|
|
|
2628 |
apply(simp_all)[5]
|
|
|
2629 |
apply(clarify)
|
|
|
2630 |
apply(case_tac "v1a = v1b")
|
|
|
2631 |
apply(simp)
|
|
|
2632 |
apply(rule ValOrd.intros(1))
|
|
|
2633 |
apply (metis ValOrd.intros(1))
|
|
|
2634 |
apply(rule ValOrd.intros(2))
|
|
|
2635 |
apply(auto)[2]
|
|
|
2636 |
apply(erule contrapos_np)
|
|
|
2637 |
apply(rule ValOrd.intros(2))
|
|
|
2638 |
apply(auto)
|
|
|
2639 |
apply(erule Prf.cases)
|
|
|
2640 |
apply(simp_all)[5]
|
|
|
2641 |
apply(erule Prf.cases)
|
|
|
2642 |
apply(simp_all)[5]
|
|
|
2643 |
apply (metis Ord1 Ord3 Prf.intros(2) ValOrd2.intros(6))
|
|
|
2644 |
apply(rule ValOrd.intros)
|
|
|
2645 |
apply(erule contrapos_np)
|
|
|
2646 |
apply(rule ValOrd.intros)
|
|
|
2647 |
apply (metis le_eq_less_or_eq neq_iff)
|
|
|
2648 |
apply(erule Prf.cases)
|
|
|
2649 |
apply(simp_all)[5]
|
|
|
2650 |
apply(rule ValOrd.intros)
|
|
|
2651 |
apply(erule contrapos_np)
|
|
|
2652 |
apply(rule ValOrd.intros)
|
|
|
2653 |
apply (metis le_eq_less_or_eq neq_iff)
|
|
|
2654 |
apply(rule ValOrd.intros)
|
|
|
2655 |
apply(erule contrapos_np)
|
|
|
2656 |
apply(rule ValOrd.intros)
|
|
|
2657 |
by metis
|
|
|
2658 |
|
|
|
2659 |
lemma ValOrd_anti:
|
|
|
2660 |
shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r; v1 \<succ>r v2; v2 \<succ>r v1\<rbrakk> \<Longrightarrow> v1 = v2"
|
|
|
2661 |
apply(induct r arbitrary: v1 v2)
|
|
|
2662 |
apply(erule Prf.cases)
|
|
|
2663 |
apply(simp_all)[5]
|
|
|
2664 |
apply(erule Prf.cases)
|
|
|
2665 |
apply(simp_all)[5]
|
|
|
2666 |
apply(erule Prf.cases)
|
|
|
2667 |
apply(simp_all)[5]
|
|
|
2668 |
apply(erule Prf.cases)
|
|
|
2669 |
apply(simp_all)[5]
|
|
|
2670 |
apply(erule Prf.cases)
|
|
|
2671 |
apply(simp_all)[5]
|
|
|
2672 |
apply(erule Prf.cases)
|
|
|
2673 |
apply(simp_all)[5]
|
|
|
2674 |
apply(erule Prf.cases)
|
|
|
2675 |
apply(simp_all)[5]
|
|
|
2676 |
apply(erule ValOrd.cases)
|
|
|
2677 |
apply(simp_all)[8]
|
|
|
2678 |
apply(erule ValOrd.cases)
|
|
|
2679 |
apply(simp_all)[8]
|
|
|
2680 |
apply(erule ValOrd.cases)
|
|
|
2681 |
apply(simp_all)[8]
|
|
|
2682 |
apply(erule Prf.cases)
|
|
|
2683 |
apply(simp_all)[5]
|
|
|
2684 |
apply(erule Prf.cases)
|
|
|
2685 |
apply(simp_all)[5]
|
|
|
2686 |
apply(erule ValOrd.cases)
|
|
|
2687 |
apply(simp_all)[8]
|
|
|
2688 |
apply(erule ValOrd.cases)
|
|
|
2689 |
apply(simp_all)[8]
|
|
|
2690 |
apply(erule ValOrd.cases)
|
|
|
2691 |
apply(simp_all)[8]
|
|
|
2692 |
apply(erule ValOrd.cases)
|
|
|
2693 |
apply(simp_all)[8]
|
|
|
2694 |
apply(erule Prf.cases)
|
|
|
2695 |
apply(simp_all)[5]
|
|
|
2696 |
apply(erule ValOrd.cases)
|
|
|
2697 |
apply(simp_all)[8]
|
|
|
2698 |
apply(erule ValOrd.cases)
|
|
|
2699 |
apply(simp_all)[8]
|
|
|
2700 |
apply(erule ValOrd.cases)
|
|
|
2701 |
apply(simp_all)[8]
|
|
|
2702 |
apply(erule ValOrd.cases)
|
|
|
2703 |
apply(simp_all)[8]
|
|
|
2704 |
done
|
|
|
2705 |
|
|
|
2706 |
lemma POSIX_ALT_I1:
|
|
|
2707 |
assumes "POSIX v1 r1"
|
|
|
2708 |
shows "POSIX (Left v1) (ALT r1 r2)"
|
|
|
2709 |
using assms
|
|
|
2710 |
unfolding POSIX_def
|
|
|
2711 |
apply(auto)
|
|
|
2712 |
apply (metis Prf.intros(2))
|
|
|
2713 |
apply(rotate_tac 2)
|
|
|
2714 |
apply(erule Prf.cases)
|
|
|
2715 |
apply(simp_all)[5]
|
|
|
2716 |
apply(auto)
|
|
|
2717 |
apply(rule ValOrd.intros)
|
|
|
2718 |
apply(auto)
|
|
|
2719 |
apply(rule ValOrd.intros)
|
|
|
2720 |
by (metis le_eq_less_or_eq length_sprefix sprefix_def)
|
|
|
2721 |
|
|
|
2722 |
lemma POSIX_ALT_I2:
|
|
|
2723 |
assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')"
|
|
|
2724 |
shows "POSIX (Right v2) (ALT r1 r2)"
|
|
|
2725 |
using assms
|
|
|
2726 |
unfolding POSIX_def
|
|
|
2727 |
apply(auto)
|
|
|
2728 |
apply (metis Prf.intros)
|
|
|
2729 |
apply(rotate_tac 3)
|
|
|
2730 |
apply(erule Prf.cases)
|
|
|
2731 |
apply(simp_all)[5]
|
|
|
2732 |
apply(auto)
|
|
|
2733 |
apply(rule ValOrd.intros)
|
|
|
2734 |
apply metis
|
|
|
2735 |
apply(rule ValOrd.intros)
|
|
|
2736 |
apply metis
|
|
|
2737 |
done
|
|
|
2738 |
|
|
|
2739 |
thm PMatch.intros[no_vars]
|
|
|
2740 |
|
|
|
2741 |
lemma POSIX_PMatch:
|
|
|
2742 |
assumes "s \<in> r \<rightarrow> v" "\<turnstile> v' : r"
|
|
|
2743 |
shows "length (flat v') \<le> length (flat v)"
|
|
|
2744 |
using assms
|
|
|
2745 |
apply(induct arbitrary: s v v' rule: rexp.induct)
|
|
|
2746 |
apply(erule Prf.cases)
|
|
|
2747 |
apply(simp_all)[5]
|
|
|
2748 |
apply(erule Prf.cases)
|
|
|
2749 |
apply(simp_all)[5]
|
|
|
2750 |
apply(erule Prf.cases)
|
|
|
2751 |
apply(simp_all)[5]
|
|
|
2752 |
apply(erule PMatch.cases)
|
|
|
2753 |
apply(simp_all)[5]
|
|
|
2754 |
defer
|
|
|
2755 |
apply(erule Prf.cases)
|
|
|
2756 |
apply(simp_all)[5]
|
|
|
2757 |
apply(erule PMatch.cases)
|
|
|
2758 |
apply(simp_all)[5]
|
|
|
2759 |
apply(clarify)
|
|
|
2760 |
apply(simp add: L_flat_Prf)
|
|
|
2761 |
|
|
|
2762 |
apply(clarify)
|
|
|
2763 |
apply (metis ValOrd.intros(8))
|
|
|
2764 |
apply (metis POSIX_ALT_I1)
|
|
|
2765 |
apply(rule POSIX_ALT_I2)
|
|
|
2766 |
apply(simp)
|
|
|
2767 |
apply(auto)[1]
|
|
|
2768 |
apply(simp add: POSIX_def)
|
|
|
2769 |
apply(frule PMatch1(1))
|
|
|
2770 |
apply(frule PMatch1(2))
|
|
|
2771 |
apply(simp)
|
|
|
2772 |
|
|
|
2773 |
|
|
|
2774 |
lemma POSIX_PMatch:
|
|
|
2775 |
assumes "s \<in> r \<rightarrow> v"
|
|
|
2776 |
shows "POSIX v r"
|
|
|
2777 |
using assms
|
|
|
2778 |
apply(induct arbitrary: rule: PMatch.induct)
|
|
|
2779 |
apply(auto)
|
|
|
2780 |
apply(simp add: POSIX_def)
|
|
|
2781 |
apply(auto)[1]
|
|
|
2782 |
apply (metis Prf.intros(4))
|
|
|
2783 |
apply(erule Prf.cases)
|
|
|
2784 |
apply(simp_all)[5]
|
|
|
2785 |
apply (metis ValOrd.intros(7))
|
|
|
2786 |
apply(simp add: POSIX_def)
|
|
|
2787 |
apply(auto)[1]
|
|
|
2788 |
apply (metis Prf.intros(5))
|
|
|
2789 |
apply(erule Prf.cases)
|
|
|
2790 |
apply(simp_all)[5]
|
|
|
2791 |
apply (metis ValOrd.intros(8))
|
|
|
2792 |
apply (metis POSIX_ALT_I1)
|
|
|
2793 |
apply(rule POSIX_ALT_I2)
|
|
|
2794 |
apply(simp)
|
|
|
2795 |
apply(auto)[1]
|
|
|
2796 |
apply(simp add: POSIX_def)
|
|
|
2797 |
apply(frule PMatch1(1))
|
|
|
2798 |
apply(frule PMatch1(2))
|
|
|
2799 |
apply(simp)
|
|
|
2800 |
|
|
|
2801 |
|
|
|
2802 |
|
|
|
2803 |
lemma ValOrd_PMatch:
|
|
|
2804 |
assumes "s \<in> r \<rightarrow> v1" "\<turnstile> v2 : r" "flat v2 = s"
|
|
|
2805 |
shows "v1 \<succ>r v2"
|
|
|
2806 |
using assms
|
|
|
2807 |
apply(induct arbitrary: v2 rule: PMatch.induct)
|
|
|
2808 |
apply(erule Prf.cases)
|
|
|
2809 |
apply(simp_all)[5]
|
|
|
2810 |
apply (metis ValOrd.intros(7))
|
|
|
2811 |
apply(erule Prf.cases)
|
|
|
2812 |
apply(simp_all)[5]
|
|
|
2813 |
apply (metis ValOrd.intros(8))
|
|
|
2814 |
apply(erule Prf.cases)
|
|
|
2815 |
apply(simp_all)[5]
|
|
|
2816 |
apply(clarify)
|
|
|
2817 |
apply (metis ValOrd.intros(6))
|
|
|
2818 |
apply(clarify)
|
|
|
2819 |
apply (metis PMatch1(2) ValOrd.intros(3) order_refl)
|
|
|
2820 |
apply(erule Prf.cases)
|
|
|
2821 |
apply(simp_all)[5]
|
|
|
2822 |
apply(clarify)
|
|
|
2823 |
apply (metis Prf_flat_L)
|
|
|
2824 |
apply (metis ValOrd.intros(5))
|
|
|
2825 |
(* Seq case *)
|
|
|
2826 |
apply(erule Prf.cases)
|
|
|
2827 |
apply(simp_all)[5]
|
|
|
2828 |
apply(auto)
|
|
|
2829 |
apply(case_tac "v1 = v1a")
|
|
|
2830 |
apply(auto)
|
|
|
2831 |
apply (metis PMatch1(2) ValOrd.intros(1) same_append_eq)
|
|
|
2832 |
apply(rule ValOrd.intros(2))
|
|
|
2833 |
apply(auto)
|
|
|
2834 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
2835 |
apply(drule_tac meta_mp)
|
|
|
2836 |
apply(simp)
|
|
|
2837 |
apply(drule_tac meta_mp)
|
|
|
2838 |
prefer 2
|
|
|
2839 |
apply(simp)
|
|
|
2840 |
apply(simp add: append_eq_append_conv2)
|
|
|
2841 |
apply(auto)
|
|
|
2842 |
apply (metis Prf_flat_L)
|
|
|
2843 |
apply(case_tac "us = []")
|
|
|
2844 |
apply(simp)
|
|
|
2845 |
apply(drule_tac x="us" in spec)
|
|
|
2846 |
apply(drule mp)
|
|
|
2847 |
|
|
|
2848 |
thm L_flat_Prf
|
|
|
2849 |
apply(simp add: L_flat_Prf)
|
|
|
2850 |
thm append_eq_append_conv2
|
|
|
2851 |
apply(simp add: append_eq_append_conv2)
|
|
|
2852 |
apply(auto)
|
|
|
2853 |
apply(drule_tac x="us" in spec)
|
|
|
2854 |
apply(drule mp)
|
|
|
2855 |
apply metis
|
|
|
2856 |
apply (metis append_Nil2)
|
|
|
2857 |
apply(case_tac "us = []")
|
|
|
2858 |
apply(auto)
|
|
|
2859 |
apply(drule_tac x="s2" in spec)
|
|
|
2860 |
apply(drule mp)
|
|
|
2861 |
|
|
|
2862 |
apply(auto)[1]
|
|
|
2863 |
apply(drule_tac x="v1a" in meta_spec)
|
|
|
2864 |
apply(simp)
|
|
|
2865 |
|
|
|
2866 |
lemma refl_on_ValOrd:
|
|
|
2867 |
"refl_on (Values r s) {(v1, v2). v1 \<succ>r v2 \<and> v1 \<in> Values r s \<and> v2 \<in> Values r s}"
|
|
|
2868 |
unfolding refl_on_def
|
|
|
2869 |
apply(auto)
|
|
|
2870 |
apply(rule ValOrd_refl)
|
|
|
2871 |
apply(simp add: Values_def)
|
|
|
2872 |
done
|
|
|
2873 |
|
|
|
2874 |
|
|
|
2875 |
section {* Posix definition *}
|
|
|
2876 |
|
|
|
2877 |
definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
2878 |
where
|
|
|
2879 |
"POSIX v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v \<succ>r v'))"
|
|
|
2880 |
|
|
|
2881 |
definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> bool"
|
|
|
2882 |
where
|
|
|
2883 |
"POSIX2 v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v 2\<succ> v'))"
|
|
|
2884 |
|
|
|
2885 |
lemma "POSIX v r = POSIX2 v r"
|
|
|
2886 |
unfolding POSIX_def POSIX2_def
|
|
|
2887 |
apply(auto)
|
|
|
2888 |
apply(rule Ord1)
|
|
|
2889 |
apply(auto)
|
|
|
2890 |
apply(rule Ord3)
|
|
|
2891 |
apply(auto)
|
|
|
2892 |
done
|
|
|
2893 |
|
|
|
2894 |
section {* POSIX for some constructors *}
|
|
|
2895 |
|
|
|
2896 |
lemma POSIX_SEQ1:
|
|
|
2897 |
assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2"
|
|
|
2898 |
shows "POSIX v1 r1"
|
|
|
2899 |
using assms
|
|
|
2900 |
unfolding POSIX_def
|
|
|
2901 |
apply(auto)
|
|
|
2902 |
apply(drule_tac x="Seq v' v2" in spec)
|
|
|
2903 |
apply(simp)
|
|
|
2904 |
apply(erule impE)
|
|
|
2905 |
apply(rule Prf.intros)
|
|
|
2906 |
apply(simp)
|
|
|
2907 |
apply(simp)
|
|
|
2908 |
apply(erule ValOrd.cases)
|
|
|
2909 |
apply(simp_all)
|
|
|
2910 |
apply(clarify)
|
|
|
2911 |
by (metis ValOrd_refl)
|
|
|
2912 |
|
|
|
2913 |
lemma POSIX_SEQ2:
|
|
|
2914 |
assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2"
|
|
|
2915 |
shows "POSIX v2 r2"
|
|
|
2916 |
using assms
|
|
|
2917 |
unfolding POSIX_def
|
|
|
2918 |
apply(auto)
|
|
|
2919 |
apply(drule_tac x="Seq v1 v'" in spec)
|
|
|
2920 |
apply(simp)
|
|
|
2921 |
apply(erule impE)
|
|
|
2922 |
apply(rule Prf.intros)
|
|
|
2923 |
apply(simp)
|
|
|
2924 |
apply(simp)
|
|
|
2925 |
apply(erule ValOrd.cases)
|
|
|
2926 |
apply(simp_all)
|
|
|
2927 |
done
|
|
|
2928 |
|
|
|
2929 |
lemma POSIX_ALT2:
|
|
|
2930 |
assumes "POSIX (Left v1) (ALT r1 r2)"
|
|
|
2931 |
shows "POSIX v1 r1"
|
|
|
2932 |
using assms
|
|
|
2933 |
unfolding POSIX_def
|
|
|
2934 |
apply(auto)
|
|
|
2935 |
apply(erule Prf.cases)
|
|
|
2936 |
apply(simp_all)[5]
|
|
|
2937 |
apply(drule_tac x="Left v'" in spec)
|
|
|
2938 |
apply(simp)
|
|
|
2939 |
apply(drule mp)
|
|
|
2940 |
apply(rule Prf.intros)
|
|
|
2941 |
apply(auto)
|
|
|
2942 |
apply(erule ValOrd.cases)
|
|
|
2943 |
apply(simp_all)
|
|
|
2944 |
done
|
|
|
2945 |
|
|
|
2946 |
lemma POSIX_ALT1a:
|
|
|
2947 |
assumes "POSIX (Right v2) (ALT r1 r2)"
|
|
|
2948 |
shows "POSIX v2 r2"
|
|
|
2949 |
using assms
|
|
|
2950 |
unfolding POSIX_def
|
|
|
2951 |
apply(auto)
|
|
|
2952 |
apply(erule Prf.cases)
|
|
|
2953 |
apply(simp_all)[5]
|
|
|
2954 |
apply(drule_tac x="Right v'" in spec)
|
|
|
2955 |
apply(simp)
|
|
|
2956 |
apply(drule mp)
|
|
|
2957 |
apply(rule Prf.intros)
|
|
|
2958 |
apply(auto)
|
|
|
2959 |
apply(erule ValOrd.cases)
|
|
|
2960 |
apply(simp_all)
|
|
|
2961 |
done
|
|
|
2962 |
|
|
|
2963 |
lemma POSIX_ALT1b:
|
|
|
2964 |
assumes "POSIX (Right v2) (ALT r1 r2)"
|
|
|
2965 |
shows "(\<forall>v'. (\<turnstile> v' : r2 \<and> flat v' = flat v2) \<longrightarrow> v2 \<succ>r2 v')"
|
|
|
2966 |
using assms
|
|
|
2967 |
apply(drule_tac POSIX_ALT1a)
|
|
|
2968 |
unfolding POSIX_def
|
|
|
2969 |
apply(auto)
|
|
|
2970 |
done
|
|
|
2971 |
|
|
|
2972 |
lemma POSIX_ALT_I1:
|
|
|
2973 |
assumes "POSIX v1 r1"
|
|
|
2974 |
shows "POSIX (Left v1) (ALT r1 r2)"
|
|
|
2975 |
using assms
|
|
|
2976 |
unfolding POSIX_def
|
|
|
2977 |
apply(auto)
|
|
|
2978 |
apply (metis Prf.intros(2))
|
|
|
2979 |
apply(rotate_tac 2)
|
|
|
2980 |
apply(erule Prf.cases)
|
|
|
2981 |
apply(simp_all)[5]
|
|
|
2982 |
apply(auto)
|
|
|
2983 |
apply(rule ValOrd.intros)
|
|
|
2984 |
apply(auto)
|
|
|
2985 |
apply(rule ValOrd.intros)
|
|
|
2986 |
by simp
|
|
|
2987 |
|
|
|
2988 |
lemma POSIX_ALT_I2:
|
|
|
2989 |
assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')"
|
|
|
2990 |
shows "POSIX (Right v2) (ALT r1 r2)"
|
|
|
2991 |
using assms
|
|
|
2992 |
unfolding POSIX_def
|
|
|
2993 |
apply(auto)
|
|
|
2994 |
apply (metis Prf.intros)
|
|
|
2995 |
apply(rotate_tac 3)
|
|
|
2996 |
apply(erule Prf.cases)
|
|
|
2997 |
apply(simp_all)[5]
|
|
|
2998 |
apply(auto)
|
|
|
2999 |
apply(rule ValOrd.intros)
|
|
|
3000 |
apply metis
|
|
|
3001 |
done
|
|
|
3002 |
|
|
|
3003 |
lemma mkeps_POSIX:
|
|
|
3004 |
assumes "nullable r"
|
|
|
3005 |
shows "POSIX (mkeps r) r"
|
|
|
3006 |
using assms
|
|
|
3007 |
apply(induct r)
|
|
|
3008 |
apply(auto)[1]
|
|
|
3009 |
apply(simp add: POSIX_def)
|
|
|
3010 |
apply(auto)[1]
|
|
|
3011 |
apply (metis Prf.intros(4))
|
|
|
3012 |
apply(erule Prf.cases)
|
|
|
3013 |
apply(simp_all)[5]
|
|
|
3014 |
apply (metis ValOrd.intros)
|
|
|
3015 |
apply(simp)
|
|
|
3016 |
apply(auto)[1]
|
|
|
3017 |
apply(simp add: POSIX_def)
|
|
|
3018 |
apply(auto)[1]
|
|
|
3019 |
apply (metis mkeps.simps(2) mkeps_nullable nullable.simps(5))
|
|
|
3020 |
apply(rotate_tac 6)
|
|
|
3021 |
apply(erule Prf.cases)
|
|
|
3022 |
apply(simp_all)[5]
|
|
|
3023 |
apply (simp add: mkeps_flat)
|
|
|
3024 |
apply(case_tac "mkeps r1a = v1")
|
|
|
3025 |
apply(simp)
|
|
|
3026 |
apply (metis ValOrd.intros(1))
|
|
|
3027 |
apply (rule ValOrd.intros(2))
|
|
|
3028 |
apply metis
|
|
|
3029 |
apply(simp)
|
|
|
3030 |
(* ALT case *)
|
|
|
3031 |
thm mkeps.simps
|
|
|
3032 |
apply(simp)
|
|
|
3033 |
apply(erule disjE)
|
|
|
3034 |
apply(simp)
|
|
|
3035 |
apply (metis POSIX_ALT_I1)
|
|
|
3036 |
(* *)
|
|
|
3037 |
apply(auto)[1]
|
|
|
3038 |
thm POSIX_ALT_I1
|
|
|
3039 |
apply (metis POSIX_ALT_I1)
|
|
|
3040 |
apply(simp (no_asm) add: POSIX_def)
|
|
|
3041 |
apply(auto)[1]
|
|
|
3042 |
apply(rule Prf.intros(3))
|
|
|
3043 |
apply(simp only: POSIX_def)
|
|
|
3044 |
apply(rotate_tac 4)
|
|
|
3045 |
apply(erule Prf.cases)
|
|
|
3046 |
apply(simp_all)[5]
|
|
|
3047 |
thm mkeps_flat
|
|
|
3048 |
apply(simp add: mkeps_flat)
|
|
|
3049 |
apply(auto)[1]
|
|
|
3050 |
thm Prf_flat_L nullable_correctness
|
|
|
3051 |
apply (metis Prf_flat_L nullable_correctness)
|
|
|
3052 |
apply(rule ValOrd.intros)
|
|
|
3053 |
apply(subst (asm) POSIX_def)
|
|
|
3054 |
apply(clarify)
|
|
|
3055 |
apply(drule_tac x="v2" in spec)
|
|
|
3056 |
by simp
|
|
|
3057 |
|
|
|
3058 |
|
|
|
3059 |
|
|
|
3060 |
text {*
|
|
|
3061 |
Injection value is related to r
|
|
|
3062 |
*}
|
|
|
3063 |
|
|
|
3064 |
|
|
|
3065 |
|
|
|
3066 |
text {*
|
|
|
3067 |
The string behind the injection value is an added c
|
|
|
3068 |
*}
|
|
|
3069 |
|
|
|
3070 |
|
|
|
3071 |
lemma injval_inj: "inj_on (injval r c) {v. \<turnstile> v : der c r}"
|
|
|
3072 |
apply(induct c r rule: der.induct)
|
|
|
3073 |
unfolding inj_on_def
|
|
|
3074 |
apply(auto)[1]
|
|
|
3075 |
apply(erule Prf.cases)
|
|
|
3076 |
apply(simp_all)[5]
|
|
|
3077 |
apply(auto)[1]
|
|
|
3078 |
apply(erule Prf.cases)
|
|
|
3079 |
apply(simp_all)[5]
|
|
|
3080 |
apply(auto)[1]
|
|
|
3081 |
apply(erule Prf.cases)
|
|
|
3082 |
apply(simp_all)[5]
|
|
|
3083 |
apply(erule Prf.cases)
|
|
|
3084 |
apply(simp_all)[5]
|
|
|
3085 |
apply(erule Prf.cases)
|
|
|
3086 |
apply(simp_all)[5]
|
|
|
3087 |
apply(auto)[1]
|
|
|
3088 |
apply(erule Prf.cases)
|
|
|
3089 |
apply(simp_all)[5]
|
|
|
3090 |
apply(erule Prf.cases)
|
|
|
3091 |
apply(simp_all)[5]
|
|
|
3092 |
apply(erule Prf.cases)
|
|
|
3093 |
apply(simp_all)[5]
|
|
|
3094 |
apply(auto)[1]
|
|
|
3095 |
apply(erule Prf.cases)
|
|
|
3096 |
apply(simp_all)[5]
|
|
|
3097 |
apply(erule Prf.cases)
|
|
|
3098 |
apply(simp_all)[5]
|
|
|
3099 |
apply(clarify)
|
|
|
3100 |
apply(erule Prf.cases)
|
|
|
3101 |
apply(simp_all)[5]
|
|
|
3102 |
apply(erule Prf.cases)
|
|
|
3103 |
apply(simp_all)[5]
|
|
|
3104 |
apply(clarify)
|
|
|
3105 |
apply(erule Prf.cases)
|
|
|
3106 |
apply(simp_all)[5]
|
|
|
3107 |
apply(clarify)
|
|
|
3108 |
apply (metis list.distinct(1) mkeps_flat v4)
|
|
|
3109 |
apply(erule Prf.cases)
|
|
|
3110 |
apply(simp_all)[5]
|
|
|
3111 |
apply(clarify)
|
|
|
3112 |
apply(rotate_tac 6)
|
|
|
3113 |
apply(erule Prf.cases)
|
|
|
3114 |
apply(simp_all)[5]
|
|
|
3115 |
apply (metis list.distinct(1) mkeps_flat v4)
|
|
|
3116 |
apply(erule Prf.cases)
|
|
|
3117 |
apply(simp_all)[5]
|
|
|
3118 |
apply(erule Prf.cases)
|
|
|
3119 |
apply(simp_all)[5]
|
|
|
3120 |
done
|
|
|
3121 |
|
|
|
3122 |
lemma Values_nullable:
|
|
|
3123 |
assumes "nullable r1"
|
|
|
3124 |
shows "mkeps r1 \<in> Values r1 s"
|
|
|
3125 |
using assms
|
|
|
3126 |
apply(induct r1 arbitrary: s)
|
|
|
3127 |
apply(simp_all)
|
|
|
3128 |
apply(simp add: Values_recs)
|
|
|
3129 |
apply(simp add: Values_recs)
|
|
|
3130 |
apply(simp add: Values_recs)
|
|
|
3131 |
apply(auto)[1]
|
|
|
3132 |
done
|
|
|
3133 |
|
|
|
3134 |
lemma Values_injval:
|
|
|
3135 |
assumes "v \<in> Values (der c r) s"
|
|
|
3136 |
shows "injval r c v \<in> Values r (c#s)"
|
|
|
3137 |
using assms
|
|
|
3138 |
apply(induct c r arbitrary: v s rule: der.induct)
|
|
|
3139 |
apply(simp add: Values_recs)
|
|
|
3140 |
apply(simp add: Values_recs)
|
|
|
3141 |
apply(case_tac "c = c'")
|
|
|
3142 |
apply(simp)
|
|
|
3143 |
apply(simp add: Values_recs)
|
|
|
3144 |
apply(simp add: prefix_def)
|
|
|
3145 |
apply(simp)
|
|
|
3146 |
apply(simp add: Values_recs)
|
|
|
3147 |
apply(simp)
|
|
|
3148 |
apply(simp add: Values_recs)
|
|
|
3149 |
apply(auto)[1]
|
|
|
3150 |
apply(case_tac "nullable r1")
|
|
|
3151 |
apply(simp)
|
|
|
3152 |
apply(simp add: Values_recs)
|
|
|
3153 |
apply(auto)[1]
|
|
|
3154 |
apply(simp add: rest_def)
|
|
|
3155 |
apply(subst v4)
|
|
|
3156 |
apply(simp add: Values_def)
|
|
|
3157 |
apply(simp add: Values_def)
|
|
|
3158 |
apply(rule Values_nullable)
|
|
|
3159 |
apply(assumption)
|
|
|
3160 |
apply(simp add: rest_def)
|
|
|
3161 |
apply(subst mkeps_flat)
|
|
|
3162 |
apply(assumption)
|
|
|
3163 |
apply(simp)
|
|
|
3164 |
apply(simp)
|
|
|
3165 |
apply(simp add: Values_recs)
|
|
|
3166 |
apply(auto)[1]
|
|
|
3167 |
apply(simp add: rest_def)
|
|
|
3168 |
apply(subst v4)
|
|
|
3169 |
apply(simp add: Values_def)
|
|
|
3170 |
apply(simp add: Values_def)
|
|
|
3171 |
done
|
|
|
3172 |
|
|
|
3173 |
lemma Values_projval:
|
|
|
3174 |
assumes "v \<in> Values r (c#s)" "\<exists>s. flat v = c # s"
|
|
|
3175 |
shows "projval r c v \<in> Values (der c r) s"
|
|
|
3176 |
using assms
|
|
|
3177 |
apply(induct r arbitrary: v s c rule: rexp.induct)
|
|
|
3178 |
apply(simp add: Values_recs)
|
|
|
3179 |
apply(simp add: Values_recs)
|
|
|
3180 |
apply(case_tac "c = char")
|
|
|
3181 |
apply(simp)
|
|
|
3182 |
apply(simp add: Values_recs)
|
|
|
3183 |
apply(simp)
|
|
|
3184 |
apply(simp add: Values_recs)
|
|
|
3185 |
apply(simp add: prefix_def)
|
|
|
3186 |
apply(case_tac "nullable rexp1")
|
|
|
3187 |
apply(simp)
|
|
|
3188 |
apply(simp add: Values_recs)
|
|
|
3189 |
apply(auto)[1]
|
|
|
3190 |
apply(simp add: rest_def)
|
|
|
3191 |
apply (metis hd_Cons_tl hd_append2 list.sel(1))
|
|
|
3192 |
apply(simp add: rest_def)
|
|
|
3193 |
apply(simp add: append_eq_Cons_conv)
|
|
|
3194 |
apply(auto)[1]
|
|
|
3195 |
apply(subst v4_proj2)
|
|
|
3196 |
apply(simp add: Values_def)
|
|
|
3197 |
apply(assumption)
|
|
|
3198 |
apply(simp)
|
|
|
3199 |
apply(simp)
|
|
|
3200 |
apply(simp add: Values_recs)
|
|
|
3201 |
apply(auto)[1]
|
|
|
3202 |
apply(auto simp add: Values_def not_nullable_flat)[1]
|
|
|
3203 |
apply(simp add: append_eq_Cons_conv)
|
|
|
3204 |
apply(auto)[1]
|
|
|
3205 |
apply(simp add: append_eq_Cons_conv)
|
|
|
3206 |
apply(auto)[1]
|
|
|
3207 |
apply(simp add: rest_def)
|
|
|
3208 |
apply(subst v4_proj2)
|
|
|
3209 |
apply(simp add: Values_def)
|
|
|
3210 |
apply(assumption)
|
|
|
3211 |
apply(simp)
|
|
|
3212 |
apply(simp add: Values_recs)
|
|
|
3213 |
apply(auto)[1]
|
|
|
3214 |
done
|
|
|
3215 |
|
|
|
3216 |
|
|
|
3217 |
definition "MValue v r s \<equiv> (v \<in> Values r s \<and> (\<forall>v' \<in> Values r s. v 2\<succ> v'))"
|
|
|
3218 |
|
|
|
3219 |
lemma MValue_ALTE:
|
|
|
3220 |
assumes "MValue v (ALT r1 r2) s"
|
|
|
3221 |
shows "(\<exists>vl. v = Left vl \<and> MValue vl r1 s \<and> (\<forall>vr \<in> Values r2 s. length (flat vr) \<le> length (flat vl))) \<or>
|
|
|
3222 |
(\<exists>vr. v = Right vr \<and> MValue vr r2 s \<and> (\<forall>vl \<in> Values r1 s. length (flat vl) < length (flat vr)))"
|
|
|
3223 |
using assms
|
|
|
3224 |
apply(simp add: MValue_def)
|
|
|
3225 |
apply(simp add: Values_recs)
|
|
|
3226 |
apply(auto)
|
|
|
3227 |
apply(drule_tac x="Left x" in bspec)
|
|
|
3228 |
apply(simp)
|
|
|
3229 |
apply(erule ValOrd2.cases)
|
|
|
3230 |
apply(simp_all)
|
|
|
3231 |
apply(drule_tac x="Right vr" in bspec)
|
|
|
3232 |
apply(simp)
|
|
|
3233 |
apply(erule ValOrd2.cases)
|
|
|
3234 |
apply(simp_all)
|
|
|
3235 |
apply(drule_tac x="Right x" in bspec)
|
|
|
3236 |
apply(simp)
|
|
|
3237 |
apply(erule ValOrd2.cases)
|
|
|
3238 |
apply(simp_all)
|
|
|
3239 |
apply(drule_tac x="Left vl" in bspec)
|
|
|
3240 |
apply(simp)
|
|
|
3241 |
apply(erule ValOrd2.cases)
|
|
|
3242 |
apply(simp_all)
|
|
|
3243 |
done
|
|
|
3244 |
|
|
|
3245 |
lemma MValue_ALTI1:
|
|
|
3246 |
assumes "MValue vl r1 s" "\<forall>vr \<in> Values r2 s. length (flat vr) \<le> length (flat vl)"
|
|
|
3247 |
shows "MValue (Left vl) (ALT r1 r2) s"
|
|
|
3248 |
using assms
|
|
|
3249 |
apply(simp add: MValue_def)
|
|
|
3250 |
apply(simp add: Values_recs)
|
|
|
3251 |
apply(auto)
|
|
|
3252 |
apply(rule ValOrd2.intros)
|
|
|
3253 |
apply metis
|
|
|
3254 |
apply(rule ValOrd2.intros)
|
|
|
3255 |
apply metis
|
|
|
3256 |
done
|
|
|
3257 |
|
|
|
3258 |
lemma MValue_ALTI2:
|
|
|
3259 |
assumes "MValue vr r2 s" "\<forall>vl \<in> Values r1 s. length (flat vl) < length (flat vr)"
|
|
|
3260 |
shows "MValue (Right vr) (ALT r1 r2) s"
|
|
|
3261 |
using assms
|
|
|
3262 |
apply(simp add: MValue_def)
|
|
|
3263 |
apply(simp add: Values_recs)
|
|
|
3264 |
apply(auto)
|
|
|
3265 |
apply(rule ValOrd2.intros)
|
|
|
3266 |
apply metis
|
|
|
3267 |
apply(rule ValOrd2.intros)
|
|
|
3268 |
apply metis
|
|
|
3269 |
done
|
|
|
3270 |
|
|
|
3271 |
lemma t: "(c#xs = c#ys) \<Longrightarrow> xs = ys"
|
|
|
3272 |
by (metis list.sel(3))
|
|
|
3273 |
|
|
|
3274 |
lemma t2: "(xs = ys) \<Longrightarrow> (c#xs) = (c#ys)"
|
|
|
3275 |
by (metis)
|
|
|
3276 |
|
|
|
3277 |
lemma "\<not>(nullable r) \<Longrightarrow> \<not>(\<exists>v. \<turnstile> v : r \<and> flat v = [])"
|
|
|
3278 |
by (metis Prf_flat_L nullable_correctness)
|
|
|
3279 |
|
|
|
3280 |
|
|
|
3281 |
lemma LeftRight:
|
|
|
3282 |
assumes "(Left v1) \<succ>(der c (ALT r1 r2)) (Right v2)"
|
|
|
3283 |
and "\<turnstile> v1 : der c r1" "\<turnstile> v2 : der c r2"
|
|
|
3284 |
shows "(injval (ALT r1 r2) c (Left v1)) \<succ>(ALT r1 r2) (injval (ALT r1 r2) c (Right v2))"
|
|
|
3285 |
using assms
|
|
|
3286 |
apply(simp)
|
|
|
3287 |
apply(erule ValOrd.cases)
|
|
|
3288 |
apply(simp_all)[8]
|
|
|
3289 |
apply(rule ValOrd.intros)
|
|
|
3290 |
apply(clarify)
|
|
|
3291 |
apply(subst v4)
|
|
|
3292 |
apply(simp)
|
|
|
3293 |
apply(subst v4)
|
|
|
3294 |
apply(simp)
|
|
|
3295 |
apply(simp)
|
|
|
3296 |
done
|
|
|
3297 |
|
|
|
3298 |
lemma RightLeft:
|
|
|
3299 |
assumes "(Right v1) \<succ>(der c (ALT r1 r2)) (Left v2)"
|
|
|
3300 |
and "\<turnstile> v1 : der c r2" "\<turnstile> v2 : der c r1"
|
|
|
3301 |
shows "(injval (ALT r1 r2) c (Right v1)) \<succ>(ALT r1 r2) (injval (ALT r1 r2) c (Left v2))"
|
|
|
3302 |
using assms
|
|
|
3303 |
apply(simp)
|
|
|
3304 |
apply(erule ValOrd.cases)
|
|
|
3305 |
apply(simp_all)[8]
|
|
|
3306 |
apply(rule ValOrd.intros)
|
|
|
3307 |
apply(clarify)
|
|
|
3308 |
apply(subst v4)
|
|
|
3309 |
apply(simp)
|
|
|
3310 |
apply(subst v4)
|
|
|
3311 |
apply(simp)
|
|
|
3312 |
apply(simp)
|
|
|
3313 |
done
|
|
|
3314 |
|
|
|
3315 |
lemma h:
|
|
|
3316 |
assumes "nullable r1" "\<turnstile> v1 : der c r1"
|
|
|
3317 |
shows "injval r1 c v1 \<succ>r1 mkeps r1"
|
|
|
3318 |
using assms
|
|
|
3319 |
apply(induct r1 arbitrary: v1 rule: der.induct)
|
|
|
3320 |
apply(simp)
|
|
|
3321 |
apply(simp)
|
|
|
3322 |
apply(erule Prf.cases)
|
|
|
3323 |
apply(simp_all)[5]
|
|
|
3324 |
apply(simp)
|
|
|
3325 |
apply(simp)
|
|
|
3326 |
apply(erule Prf.cases)
|
|
|
3327 |
apply(simp_all)[5]
|
|
|
3328 |
apply(clarify)
|
|
|
3329 |
apply(auto)[1]
|
|
|
3330 |
apply (metis ValOrd.intros(6))
|
|
|
3331 |
apply (metis ValOrd.intros(6))
|
|
|
3332 |
apply (metis ValOrd.intros(3) le_add2 list.size(3) mkeps_flat monoid_add_class.add.right_neutral)
|
|
|
3333 |
apply(auto)[1]
|
|
|
3334 |
apply (metis ValOrd.intros(4) length_greater_0_conv list.distinct(1) list.size(3) mkeps_flat v4)
|
|
|
3335 |
apply (metis ValOrd.intros(4) length_greater_0_conv list.distinct(1) list.size(3) mkeps_flat v4)
|
|
|
3336 |
apply (metis ValOrd.intros(5))
|
|
|
3337 |
apply(simp)
|
|
|
3338 |
apply(erule Prf.cases)
|
|
|
3339 |
apply(simp_all)[5]
|
|
|
3340 |
apply(clarify)
|
|
|
3341 |
apply(erule Prf.cases)
|
|
|
3342 |
apply(simp_all)[5]
|
|
|
3343 |
apply(clarify)
|
|
|
3344 |
apply (metis ValOrd.intros(2) list.distinct(1) mkeps_flat v4)
|
|
|
3345 |
apply(clarify)
|
|
|
3346 |
by (metis ValOrd.intros(1))
|
|
|
3347 |
|
|
|
3348 |
lemma LeftRightSeq:
|
|
|
3349 |
assumes "(Left (Seq v1 v2)) \<succ>(der c (SEQ r1 r2)) (Right v3)"
|
|
|
3350 |
and "nullable r1" "\<turnstile> v1 : der c r1"
|
|
|
3351 |
shows "(injval (SEQ r1 r2) c (Seq v1 v2)) \<succ>(SEQ r1 r2) (injval (SEQ r1 r2) c (Right v2))"
|
|
|
3352 |
using assms
|
|
|
3353 |
apply(simp)
|
|
|
3354 |
apply(erule ValOrd.cases)
|
|
|
3355 |
apply(simp_all)[8]
|
|
|
3356 |
apply(clarify)
|
|
|
3357 |
apply(simp)
|
|
|
3358 |
apply(rule ValOrd.intros(2))
|
|
|
3359 |
prefer 2
|
|
|
3360 |
apply (metis list.distinct(1) mkeps_flat v4)
|
|
|
3361 |
by (metis h)
|
|
|
3362 |
|
|
|
3363 |
lemma rr1:
|
|
|
3364 |
assumes "\<turnstile> v : r" "\<not>nullable r"
|
|
|
3365 |
shows "flat v \<noteq> []"
|
|
|
3366 |
using assms
|
|
|
3367 |
by (metis Prf_flat_L nullable_correctness)
|
|
|
3368 |
|
|
|
3369 |
(* HERE *)
|
|
|
3370 |
|
|
|
3371 |
lemma Prf_inj_test:
|
|
|
3372 |
assumes "v1 \<succ>(der c r) v2"
|
|
|
3373 |
"v1 \<in> Values (der c r) s"
|
|
|
3374 |
"v2 \<in> Values (der c r) s"
|
|
|
3375 |
"injval r c v1 \<in> Values r (c#s)"
|
|
|
3376 |
"injval r c v2 \<in> Values r (c#s)"
|
|
|
3377 |
shows "(injval r c v1) 2\<succ> (injval r c v2)"
|
|
|
3378 |
using assms
|
|
|
3379 |
apply(induct c r arbitrary: v1 v2 s rule: der.induct)
|
|
|
3380 |
(* NULL case *)
|
|
|
3381 |
apply(simp add: Values_recs)
|
|
|
3382 |
(* EMPTY case *)
|
|
|
3383 |
apply(simp add: Values_recs)
|
|
|
3384 |
(* CHAR case *)
|
|
|
3385 |
apply(case_tac "c = c'")
|
|
|
3386 |
apply(simp)
|
|
|
3387 |
apply(simp add: Values_recs)
|
|
|
3388 |
apply (metis ValOrd2.intros(8))
|
|
|
3389 |
apply(simp add: Values_recs)
|
|
|
3390 |
(* ALT case *)
|
|
|
3391 |
apply(simp)
|
|
|
3392 |
apply(simp add: Values_recs)
|
|
|
3393 |
apply(auto)[1]
|
|
|
3394 |
apply(erule ValOrd.cases)
|
|
|
3395 |
apply(simp_all)[8]
|
|
|
3396 |
apply (metis ValOrd2.intros(6))
|
|
|
3397 |
apply(erule ValOrd.cases)
|
|
|
3398 |
apply(simp_all)[8]
|
|
|
3399 |
apply(rule ValOrd2.intros)
|
|
|
3400 |
apply(subst v4)
|
|
|
3401 |
apply(simp add: Values_def)
|
|
|
3402 |
apply(subst v4)
|
|
|
3403 |
apply(simp add: Values_def)
|
|
|
3404 |
apply(simp)
|
|
|
3405 |
apply(erule ValOrd.cases)
|
|
|
3406 |
apply(simp_all)[8]
|
|
|
3407 |
apply(rule ValOrd2.intros)
|
|
|
3408 |
apply(subst v4)
|
|
|
3409 |
apply(simp add: Values_def)
|
|
|
3410 |
apply(subst v4)
|
|
|
3411 |
apply(simp add: Values_def)
|
|
|
3412 |
apply(simp)
|
|
|
3413 |
apply(erule ValOrd.cases)
|
|
|
3414 |
apply(simp_all)[8]
|
|
|
3415 |
apply (metis ValOrd2.intros(5))
|
|
|
3416 |
(* SEQ case*)
|
|
|
3417 |
apply(simp)
|
|
|
3418 |
apply(case_tac "nullable r1")
|
|
|
3419 |
apply(simp)
|
|
|
3420 |
defer
|
|
|
3421 |
apply(simp)
|
|
|
3422 |
apply(simp add: Values_recs)
|
|
|
3423 |
apply(auto)[1]
|
|
|
3424 |
apply(erule ValOrd.cases)
|
|
|
3425 |
apply(simp_all)[8]
|
|
|
3426 |
apply(clarify)
|
|
|
3427 |
apply(rule ValOrd2.intros)
|
|
|
3428 |
apply(simp)
|
|
|
3429 |
apply (metis Ord1)
|
|
|
3430 |
apply(clarify)
|
|
|
3431 |
apply(rule ValOrd2.intros)
|
|
|
3432 |
apply(subgoal_tac "rest v1 (flat v1 @ flat v2) = flat v2")
|
|
|
3433 |
apply(simp)
|
|
|
3434 |
apply(subgoal_tac "rest (injval r1 c v1) (c # flat v1 @ flat v2) = flat v2")
|
|
|
3435 |
apply(simp)
|
|
|
3436 |
oops
|
|
|
3437 |
|
|
|
3438 |
lemma Prf_inj_test:
|
|
|
3439 |
assumes "v1 \<succ>(der c r) v2"
|
|
|
3440 |
"v1 \<in> Values (der c r) s"
|
|
|
3441 |
"v2 \<in> Values (der c r) s"
|
|
|
3442 |
"injval r c v1 \<in> Values r (c#s)"
|
|
|
3443 |
"injval r c v2 \<in> Values r (c#s)"
|
|
|
3444 |
shows "(injval r c v1) 2\<succ> (injval r c v2)"
|
|
|
3445 |
using assms
|
|
|
3446 |
apply(induct c r arbitrary: v1 v2 s rule: der.induct)
|
|
|
3447 |
(* NULL case *)
|
|
|
3448 |
apply(simp add: Values_recs)
|
|
|
3449 |
(* EMPTY case *)
|
|
|
3450 |
apply(simp add: Values_recs)
|
|
|
3451 |
(* CHAR case *)
|
|
|
3452 |
apply(case_tac "c = c'")
|
|
|
3453 |
apply(simp)
|
|
|
3454 |
apply(simp add: Values_recs)
|
|
|
3455 |
apply (metis ValOrd2.intros(8))
|
|
|
3456 |
apply(simp add: Values_recs)
|
|
|
3457 |
(* ALT case *)
|
|
|
3458 |
apply(simp)
|
|
|
3459 |
apply(simp add: Values_recs)
|
|
|
3460 |
apply(auto)[1]
|
|
|
3461 |
apply(erule ValOrd.cases)
|
|
|
3462 |
apply(simp_all)[8]
|
|
|
3463 |
apply (metis ValOrd2.intros(6))
|
|
|
3464 |
apply(erule ValOrd.cases)
|
|
|
3465 |
apply(simp_all)[8]
|
|
|
3466 |
apply(rule ValOrd2.intros)
|
|
|
3467 |
apply(subst v4)
|
|
|
3468 |
apply(simp add: Values_def)
|
|
|
3469 |
apply(subst v4)
|
|
|
3470 |
apply(simp add: Values_def)
|
|
|
3471 |
apply(simp)
|
|
|
3472 |
apply(erule ValOrd.cases)
|
|
|
3473 |
apply(simp_all)[8]
|
|
|
3474 |
apply(rule ValOrd2.intros)
|
|
|
3475 |
apply(subst v4)
|
|
|
3476 |
apply(simp add: Values_def)
|
|
|
3477 |
apply(subst v4)
|
|
|
3478 |
apply(simp add: Values_def)
|
|
|
3479 |
apply(simp)
|
|
|
3480 |
apply(erule ValOrd.cases)
|
|
|
3481 |
apply(simp_all)[8]
|
|
|
3482 |
apply (metis ValOrd2.intros(5))
|
|
|
3483 |
(* SEQ case*)
|
|
|
3484 |
apply(simp)
|
|
|
3485 |
apply(case_tac "nullable r1")
|
|
|
3486 |
apply(simp)
|
|
|
3487 |
defer
|
|
|
3488 |
apply(simp)
|
|
|
3489 |
apply(simp add: Values_recs)
|
|
|
3490 |
apply(auto)[1]
|
|
|
3491 |
apply(erule ValOrd.cases)
|
|
|
3492 |
apply(simp_all)[8]
|
|
|
3493 |
apply(clarify)
|
|
|
3494 |
apply(rule ValOrd2.intros)
|
|
|
3495 |
apply(simp)
|
|
|
3496 |
apply (metis Ord1)
|
|
|
3497 |
apply(clarify)
|
|
|
3498 |
apply(rule ValOrd2.intros)
|
|
|
3499 |
apply metis
|
|
|
3500 |
using injval_inj
|
|
|
3501 |
apply(simp add: Values_def inj_on_def)
|
|
|
3502 |
apply metis
|
|
|
3503 |
apply(simp add: Values_recs)
|
|
|
3504 |
apply(auto)[1]
|
|
|
3505 |
apply(erule ValOrd.cases)
|
|
|
3506 |
apply(simp_all)[8]
|
|
|
3507 |
apply(clarify)
|
|
|
3508 |
apply(erule ValOrd.cases)
|
|
|
3509 |
apply(simp_all)[8]
|
|
|
3510 |
apply(clarify)
|
|
|
3511 |
apply (metis Ord1 ValOrd2.intros(1))
|
|
|
3512 |
apply(clarify)
|
|
|
3513 |
apply(rule ValOrd2.intros(2))
|
|
|
3514 |
apply metis
|
|
|
3515 |
using injval_inj
|
|
|
3516 |
apply(simp add: Values_def inj_on_def)
|
|
|
3517 |
apply metis
|
|
|
3518 |
apply(erule ValOrd.cases)
|
|
|
3519 |
apply(simp_all)[8]
|
|
|
3520 |
apply(rule ValOrd2.intros(2))
|
|
|
3521 |
thm h
|
|
|
3522 |
apply(rule Ord1)
|
|
|
3523 |
apply(rule h)
|
|
|
3524 |
apply(simp)
|
|
|
3525 |
apply(simp add: Values_def)
|
|
|
3526 |
apply(simp add: Values_def)
|
|
|
3527 |
apply (metis list.distinct(1) mkeps_flat v4)
|
|
|
3528 |
apply(erule ValOrd.cases)
|
|
|
3529 |
apply(simp_all)[8]
|
|
|
3530 |
apply(clarify)
|
|
|
3531 |
apply(simp add: Values_def)
|
|
|
3532 |
defer
|
|
|
3533 |
apply(erule ValOrd.cases)
|
|
|
3534 |
apply(simp_all)[8]
|
|
|
3535 |
apply(clarify)
|
|
|
3536 |
apply(rule ValOrd2.intros(1))
|
|
|
3537 |
apply(rotate_tac 1)
|
|
|
3538 |
apply(drule_tac x="v2" in meta_spec)
|
|
|
3539 |
apply(rotate_tac 8)
|
|
|
3540 |
apply(drule_tac x="v2'" in meta_spec)
|
|
|
3541 |
apply(rotate_tac 8)
|
|
|
3542 |
oops
|
|
|
3543 |
|
|
|
3544 |
lemma POSIX_der:
|
|
|
3545 |
assumes "POSIX v (der c r)" "\<turnstile> v : der c r"
|
|
|
3546 |
shows "POSIX (injval r c v) r"
|
|
|
3547 |
using assms
|
|
|
3548 |
unfolding POSIX_def
|
|
|
3549 |
apply(auto)
|
|
|
3550 |
thm v3
|
|
|
3551 |
apply (erule v3)
|
|
|
3552 |
thm v4
|
|
|
3553 |
apply(subst (asm) v4)
|
|
|
3554 |
apply(assumption)
|
|
|
3555 |
apply(drule_tac x="projval r c v'" in spec)
|
|
|
3556 |
apply(drule mp)
|
|
|
3557 |
apply(rule conjI)
|
|
|
3558 |
thm v3_proj
|
|
|
3559 |
apply(rule v3_proj)
|
|
|
3560 |
apply(simp)
|
|
|
3561 |
apply(rule_tac x="flat v" in exI)
|
|
|
3562 |
apply(simp)
|
|
|
3563 |
thm t
|
|
|
3564 |
apply(rule_tac c="c" in t)
|
|
|
3565 |
apply(simp)
|
|
|
3566 |
thm v4_proj
|
|
|
3567 |
apply(subst v4_proj)
|
|
|
3568 |
apply(simp)
|
|
|
3569 |
apply(rule_tac x="flat v" in exI)
|
|
|
3570 |
apply(simp)
|
|
|
3571 |
apply(simp)
|
|
|
3572 |
oops
|
|
|
3573 |
|
|
|
3574 |
lemma POSIX_der:
|
|
|
3575 |
assumes "POSIX v (der c r)" "\<turnstile> v : der c r"
|
|
|
3576 |
shows "POSIX (injval r c v) r"
|
|
|
3577 |
using assms
|
|
|
3578 |
apply(induct c r arbitrary: v rule: der.induct)
|
|
|
3579 |
(* null case*)
|
|
|
3580 |
apply(simp add: POSIX_def)
|
|
|
3581 |
apply(auto)[1]
|
|
|
3582 |
apply(erule Prf.cases)
|
|
|
3583 |
apply(simp_all)[5]
|
|
|
3584 |
apply(erule Prf.cases)
|
|
|
3585 |
apply(simp_all)[5]
|
|
|
3586 |
(* empty case *)
|
|
|
3587 |
apply(simp add: POSIX_def)
|
|
|
3588 |
apply(auto)[1]
|
|
|
3589 |
apply(erule Prf.cases)
|
|
|
3590 |
apply(simp_all)[5]
|
|
|
3591 |
apply(erule Prf.cases)
|
|
|
3592 |
apply(simp_all)[5]
|
|
|
3593 |
(* char case *)
|
|
|
3594 |
apply(simp add: POSIX_def)
|
|
|
3595 |
apply(case_tac "c = c'")
|
|
|
3596 |
apply(auto)[1]
|
|
|
3597 |
apply(erule Prf.cases)
|
|
|
3598 |
apply(simp_all)[5]
|
|
|
3599 |
apply (metis Prf.intros(5))
|
|
|
3600 |
apply(erule Prf.cases)
|
|
|
3601 |
apply(simp_all)[5]
|
|
|
3602 |
apply(erule Prf.cases)
|
|
|
3603 |
apply(simp_all)[5]
|
|
|
3604 |
apply (metis ValOrd.intros(8))
|
|
|
3605 |
apply(auto)[1]
|
|
|
3606 |
apply(erule Prf.cases)
|
|
|
3607 |
apply(simp_all)[5]
|
|
|
3608 |
apply(erule Prf.cases)
|
|
|
3609 |
apply(simp_all)[5]
|
|
|
3610 |
(* alt case *)
|
|
|
3611 |
apply(erule Prf.cases)
|
|
|
3612 |
apply(simp_all)[5]
|
|
|
3613 |
apply(clarify)
|
|
|
3614 |
apply(simp (no_asm) add: POSIX_def)
|
|
|
3615 |
apply(auto)[1]
|
|
|
3616 |
apply (metis Prf.intros(2) v3)
|
|
|
3617 |
apply(rotate_tac 4)
|
|
|
3618 |
apply(erule Prf.cases)
|
|
|
3619 |
apply(simp_all)[5]
|
|
|
3620 |
apply (metis POSIX_ALT2 POSIX_def ValOrd.intros(6))
|
|
|
3621 |
apply (metis ValOrd.intros(3) order_refl)
|
|
|
3622 |
apply(simp (no_asm) add: POSIX_def)
|
|
|
3623 |
apply(auto)[1]
|
|
|
3624 |
apply (metis Prf.intros(3) v3)
|
|
|
3625 |
apply(rotate_tac 4)
|
|
|
3626 |
apply(erule Prf.cases)
|
|
|
3627 |
apply(simp_all)[5]
|
|
|
3628 |
defer
|
|
|
3629 |
apply (metis POSIX_ALT1a POSIX_def ValOrd.intros(5))
|
|
|
3630 |
prefer 2
|
|
|
3631 |
apply(subst (asm) (5) POSIX_def)
|
|
|
3632 |
apply(auto)[1]
|
|
|
3633 |
apply(rotate_tac 5)
|
|
|
3634 |
apply(erule Prf.cases)
|
|
|
3635 |
apply(simp_all)[5]
|
|
|
3636 |
apply(rule ValOrd.intros)
|
|
|
3637 |
apply(subst (asm) v4)
|
|
|
3638 |
apply(simp)
|
|
|
3639 |
apply(drule_tac x="Left (projval r1a c v1)" in spec)
|
|
|
3640 |
apply(clarify)
|
|
|
3641 |
apply(drule mp)
|
|
|
3642 |
apply(rule conjI)
|
|
|
3643 |
apply (metis Prf.intros(2) v3_proj)
|
|
|
3644 |
apply(simp)
|
|
|
3645 |
apply (metis v4_proj2)
|
|
|
3646 |
apply(erule ValOrd.cases)
|
|
|
3647 |
apply(simp_all)[8]
|
|
|
3648 |
apply (metis less_not_refl v4_proj2)
|
|
|
3649 |
(* seq case *)
|
|
|
3650 |
apply(case_tac "nullable r1")
|
|
|
3651 |
defer
|
|
|
3652 |
apply(simp add: POSIX_def)
|
|
|
3653 |
apply(auto)[1]
|
|
|
3654 |
apply(erule Prf.cases)
|
|
|
3655 |
apply(simp_all)[5]
|
|
|
3656 |
apply (metis Prf.intros(1) v3)
|
|
|
3657 |
apply(erule Prf.cases)
|
|
|
3658 |
apply(simp_all)[5]
|
|
|
3659 |
apply(erule Prf.cases)
|
|
|
3660 |
apply(simp_all)[5]
|
|
|
3661 |
apply(clarify)
|
|
|
3662 |
apply(subst (asm) (3) v4)
|
|
|
3663 |
apply(simp)
|
|
|
3664 |
apply(simp)
|
|
|
3665 |
apply(subgoal_tac "flat v1a \<noteq> []")
|
|
|
3666 |
prefer 2
|
|
|
3667 |
apply (metis Prf_flat_L nullable_correctness)
|
|
|
3668 |
apply(subgoal_tac "\<exists>s. flat v1a = c # s")
|
|
|
3669 |
prefer 2
|
|
|
3670 |
apply (metis append_eq_Cons_conv)
|
|
|
3671 |
apply(auto)[1]
|
|
|
3672 |
oops
|
|
|
3673 |
|
|
|
3674 |
|
|
|
3675 |
lemma POSIX_ex: "\<turnstile> v : r \<Longrightarrow> \<exists>v. POSIX v r"
|
|
|
3676 |
apply(induct r arbitrary: v)
|
|
|
3677 |
apply(erule Prf.cases)
|
|
|
3678 |
apply(simp_all)[5]
|
|
|
3679 |
apply(erule Prf.cases)
|
|
|
3680 |
apply(simp_all)[5]
|
|
|
3681 |
apply(rule_tac x="Void" in exI)
|
|
|
3682 |
apply(simp add: POSIX_def)
|
|
|
3683 |
apply(auto)[1]
|
|
|
3684 |
apply (metis Prf.intros(4))
|
|
|
3685 |
apply(erule Prf.cases)
|
|
|
3686 |
apply(simp_all)[5]
|
|
|
3687 |
apply (metis ValOrd.intros(7))
|
|
|
3688 |
apply(erule Prf.cases)
|
|
|
3689 |
apply(simp_all)[5]
|
|
|
3690 |
apply(rule_tac x="Char c" in exI)
|
|
|
3691 |
apply(simp add: POSIX_def)
|
|
|
3692 |
apply(auto)[1]
|
|
|
3693 |
apply (metis Prf.intros(5))
|
|
|
3694 |
apply(erule Prf.cases)
|
|
|
3695 |
apply(simp_all)[5]
|
|
|
3696 |
apply (metis ValOrd.intros(8))
|
|
|
3697 |
apply(erule Prf.cases)
|
|
|
3698 |
apply(simp_all)[5]
|
|
|
3699 |
apply(auto)[1]
|
|
|
3700 |
apply(drule_tac x="v1" in meta_spec)
|
|
|
3701 |
apply(drule_tac x="v2" in meta_spec)
|
|
|
3702 |
apply(auto)[1]
|
|
|
3703 |
defer
|
|
|
3704 |
apply(erule Prf.cases)
|
|
|
3705 |
apply(simp_all)[5]
|
|
|
3706 |
apply(auto)[1]
|
|
|
3707 |
apply (metis POSIX_ALT_I1)
|
|
|
3708 |
apply (metis POSIX_ALT_I1 POSIX_ALT_I2)
|
|
|
3709 |
apply(case_tac "nullable r1a")
|
|
|
3710 |
apply(rule_tac x="Seq (mkeps r1a) va" in exI)
|
|
|
3711 |
apply(auto simp add: POSIX_def)[1]
|
|
|
3712 |
apply (metis Prf.intros(1) mkeps_nullable)
|
|
|
3713 |
apply(simp add: mkeps_flat)
|
|
|
3714 |
apply(rotate_tac 7)
|
|
|
3715 |
apply(erule Prf.cases)
|
|
|
3716 |
apply(simp_all)[5]
|
|
|
3717 |
apply(case_tac "mkeps r1 = v1a")
|
|
|
3718 |
apply(simp)
|
|
|
3719 |
apply (rule ValOrd.intros(1))
|
|
|
3720 |
apply (metis append_Nil mkeps_flat)
|
|
|
3721 |
apply (rule ValOrd.intros(2))
|
|
|
3722 |
apply(drule mkeps_POSIX)
|
|
|
3723 |
apply(simp add: POSIX_def)
|
|
|
3724 |
oops
|
|
|
3725 |
|
|
|
3726 |
lemma POSIX_ex2: "\<turnstile> v : r \<Longrightarrow> \<exists>v. POSIX v r \<and> \<turnstile> v : r"
|
|
|
3727 |
apply(induct r arbitrary: v)
|
|
|
3728 |
apply(erule Prf.cases)
|
|
|
3729 |
apply(simp_all)[5]
|
|
|
3730 |
apply(erule Prf.cases)
|
|
|
3731 |
apply(simp_all)[5]
|
|
|
3732 |
apply(rule_tac x="Void" in exI)
|
|
|
3733 |
apply(simp add: POSIX_def)
|
|
|
3734 |
apply(auto)[1]
|
|
|
3735 |
oops
|
|
|
3736 |
|
|
|
3737 |
lemma POSIX_ALT_cases:
|
|
|
3738 |
assumes "\<turnstile> v : (ALT r1 r2)" "POSIX v (ALT r1 r2)"
|
|
|
3739 |
shows "(\<exists>v1. v = Left v1 \<and> POSIX v1 r1) \<or> (\<exists>v2. v = Right v2 \<and> POSIX v2 r2)"
|
|
|
3740 |
using assms
|
|
|
3741 |
apply(erule_tac Prf.cases)
|
|
|
3742 |
apply(simp_all)
|
|
|
3743 |
unfolding POSIX_def
|
|
|
3744 |
apply(auto)
|
|
|
3745 |
apply (metis POSIX_ALT2 POSIX_def assms(2))
|
|
|
3746 |
by (metis POSIX_ALT1b assms(2))
|
|
|
3747 |
|
|
|
3748 |
lemma POSIX_ALT_cases2:
|
|
|
3749 |
assumes "POSIX v (ALT r1 r2)" "\<turnstile> v : (ALT r1 r2)"
|
|
|
3750 |
shows "(\<exists>v1. v = Left v1 \<and> POSIX v1 r1) \<or> (\<exists>v2. v = Right v2 \<and> POSIX v2 r2)"
|
|
|
3751 |
using assms POSIX_ALT_cases by auto
|
|
|
3752 |
|
|
|
3753 |
lemma Prf_flat_empty:
|
|
|
3754 |
assumes "\<turnstile> v : r" "flat v = []"
|
|
|
3755 |
shows "nullable r"
|
|
|
3756 |
using assms
|
|
|
3757 |
apply(induct)
|
|
|
3758 |
apply(auto)
|
|
|
3759 |
done
|
|
|
3760 |
|
|
|
3761 |
lemma POSIX_proj:
|
|
|
3762 |
assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s"
|
|
|
3763 |
shows "POSIX (projval r c v) (der c r)"
|
|
|
3764 |
using assms
|
|
|
3765 |
apply(induct r c v arbitrary: rule: projval.induct)
|
|
|
3766 |
defer
|
|
|
3767 |
defer
|
|
|
3768 |
defer
|
|
|
3769 |
defer
|
|
|
3770 |
apply(erule Prf.cases)
|
|
|
3771 |
apply(simp_all)[5]
|
|
|
3772 |
apply(erule Prf.cases)
|
|
|
3773 |
apply(simp_all)[5]
|
|
|
3774 |
apply(erule Prf.cases)
|
|
|
3775 |
apply(simp_all)[5]
|
|
|
3776 |
apply(erule Prf.cases)
|
|
|
3777 |
apply(simp_all)[5]
|
|
|
3778 |
apply(erule Prf.cases)
|
|
|
3779 |
apply(simp_all)[5]
|
|
|
3780 |
apply(erule Prf.cases)
|
|
|
3781 |
apply(simp_all)[5]
|
|
|
3782 |
apply(erule Prf.cases)
|
|
|
3783 |
apply(simp_all)[5]
|
|
|
3784 |
apply(erule Prf.cases)
|
|
|
3785 |
apply(simp_all)[5]
|
|
|
3786 |
apply(erule Prf.cases)
|
|
|
3787 |
apply(simp_all)[5]
|
|
|
3788 |
apply(erule Prf.cases)
|
|
|
3789 |
apply(simp_all)[5]
|
|
|
3790 |
apply(simp add: POSIX_def)
|
|
|
3791 |
apply(auto)[1]
|
|
|
3792 |
apply(erule Prf.cases)
|
|
|
3793 |
apply(simp_all)[5]
|
|
|
3794 |
oops
|
|
|
3795 |
|
|
|
3796 |
lemma POSIX_proj:
|
|
|
3797 |
assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s"
|
|
|
3798 |
shows "POSIX (projval r c v) (der c r)"
|
|
|
3799 |
using assms
|
|
|
3800 |
apply(induct r arbitrary: c v rule: rexp.induct)
|
|
|
3801 |
apply(erule Prf.cases)
|
|
|
3802 |
apply(simp_all)[5]
|
|
|
3803 |
apply(erule Prf.cases)
|
|
|
3804 |
apply(simp_all)[5]
|
|
|
3805 |
apply(erule Prf.cases)
|
|
|
3806 |
apply(simp_all)[5]
|
|
|
3807 |
apply(simp add: POSIX_def)
|
|
|
3808 |
apply(auto)[1]
|
|
|
3809 |
apply(erule Prf.cases)
|
|
|
3810 |
apply(simp_all)[5]
|
|
|
3811 |
oops
|
|
|
3812 |
|
|
|
3813 |
lemma POSIX_proj:
|
|
|
3814 |
assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s"
|
|
|
3815 |
shows "POSIX (projval r c v) (der c r)"
|
|
|
3816 |
using assms
|
|
|
3817 |
apply(induct r c v arbitrary: rule: projval.induct)
|
|
|
3818 |
defer
|
|
|
3819 |
defer
|
|
|
3820 |
defer
|
|
|
3821 |
defer
|
|
|
3822 |
apply(erule Prf.cases)
|
|
|
3823 |
apply(simp_all)[5]
|
|
|
3824 |
apply(erule Prf.cases)
|
|
|
3825 |
apply(simp_all)[5]
|
|
|
3826 |
apply(erule Prf.cases)
|
|
|
3827 |
apply(simp_all)[5]
|
|
|
3828 |
apply(erule Prf.cases)
|
|
|
3829 |
apply(simp_all)[5]
|
|
|
3830 |
apply(erule Prf.cases)
|
|
|
3831 |
apply(simp_all)[5]
|
|
|
3832 |
apply(erule Prf.cases)
|
|
|
3833 |
apply(simp_all)[5]
|
|
|
3834 |
apply(erule Prf.cases)
|
|
|
3835 |
apply(simp_all)[5]
|
|
|
3836 |
apply(erule Prf.cases)
|
|
|
3837 |
apply(simp_all)[5]
|
|
|
3838 |
apply(erule Prf.cases)
|
|
|
3839 |
apply(simp_all)[5]
|
|
|
3840 |
apply(erule Prf.cases)
|
|
|
3841 |
apply(simp_all)[5]
|
|
|
3842 |
apply(simp add: POSIX_def)
|
|
|
3843 |
apply(auto)[1]
|
|
|
3844 |
apply(erule Prf.cases)
|
|
|
3845 |
apply(simp_all)[5]
|
|
|
3846 |
oops
|
|
|
3847 |
|
|
|
3848 |
lemma Prf_inj:
|
|
|
3849 |
assumes "v1 \<succ>(der c r) v2" "\<turnstile> v1 : der c r" "\<turnstile> v2 : der c r" "flat v1 = flat v2"
|
|
|
3850 |
shows "(injval r c v1) \<succ>r (injval r c v2)"
|
|
|
3851 |
using assms
|
|
|
3852 |
apply(induct arbitrary: v1 v2 rule: der.induct)
|
|
|
3853 |
(* NULL case *)
|
|
|
3854 |
apply(simp)
|
|
|
3855 |
apply(erule ValOrd.cases)
|
|
|
3856 |
apply(simp_all)[8]
|
|
|
3857 |
(* EMPTY case *)
|
|
|
3858 |
apply(erule ValOrd.cases)
|
|
|
3859 |
apply(simp_all)[8]
|
|
|
3860 |
(* CHAR case *)
|
|
|
3861 |
apply(case_tac "c = c'")
|
|
|
3862 |
apply(simp)
|
|
|
3863 |
apply(erule ValOrd.cases)
|
|
|
3864 |
apply(simp_all)[8]
|
|
|
3865 |
apply(rule ValOrd.intros)
|
|
|
3866 |
apply(simp)
|
|
|
3867 |
apply(erule ValOrd.cases)
|
|
|
3868 |
apply(simp_all)[8]
|
|
|
3869 |
(* ALT case *)
|
|
|
3870 |
apply(simp)
|
|
|
3871 |
apply(erule ValOrd.cases)
|
|
|
3872 |
apply(simp_all)[8]
|
|
|
3873 |
apply(rule ValOrd.intros)
|
|
|
3874 |
apply(subst v4)
|
|
|
3875 |
apply(clarify)
|
|
|
3876 |
apply(rotate_tac 3)
|
|
|
3877 |
apply(erule Prf.cases)
|
|
|
3878 |
apply(simp_all)[5]
|
|
|
3879 |
apply(subst v4)
|
|
|
3880 |
apply(clarify)
|
|
|
3881 |
apply(rotate_tac 2)
|
|
|
3882 |
apply(erule Prf.cases)
|
|
|
3883 |
apply(simp_all)[5]
|
|
|
3884 |
apply(simp)
|
|
|
3885 |
apply(rule ValOrd.intros)
|
|
|
3886 |
apply(clarify)
|
|
|
3887 |
apply(rotate_tac 3)
|
|
|
3888 |
apply(erule Prf.cases)
|
|
|
3889 |
apply(simp_all)[5]
|
|
|
3890 |
apply(clarify)
|
|
|
3891 |
apply(erule Prf.cases)
|
|
|
3892 |
apply(simp_all)[5]
|
|
|
3893 |
apply(rule ValOrd.intros)
|
|
|
3894 |
apply(clarify)
|
|
|
3895 |
apply(erule Prf.cases)
|
|
|
3896 |
apply(simp_all)[5]
|
|
|
3897 |
apply(erule Prf.cases)
|
|
|
3898 |
apply(simp_all)[5]
|
|
|
3899 |
(* SEQ case*)
|
|
|
3900 |
apply(simp)
|
|
|
3901 |
apply(case_tac "nullable r1")
|
|
|
3902 |
defer
|
|
|
3903 |
apply(simp)
|
|
|
3904 |
apply(erule ValOrd.cases)
|
|
|
3905 |
apply(simp_all)[8]
|
|
|
3906 |
apply(clarify)
|
|
|
3907 |
apply(erule Prf.cases)
|
|
|
3908 |
apply(simp_all)[5]
|
|
|
3909 |
apply(erule Prf.cases)
|
|
|
3910 |
apply(simp_all)[5]
|
|
|
3911 |
apply(clarify)
|
|
|
3912 |
apply(rule ValOrd.intros)
|
|
|
3913 |
apply(simp)
|
|
|
3914 |
oops
|
|
|
3915 |
|
|
|
3916 |
|
|
|
3917 |
text {*
|
|
|
3918 |
Injection followed by projection is the identity.
|
|
|
3919 |
*}
|
|
|
3920 |
|
|
|
3921 |
lemma proj_inj_id:
|
|
|
3922 |
assumes "\<turnstile> v : der c r"
|
|
|
3923 |
shows "projval r c (injval r c v) = v"
|
|
|
3924 |
using assms
|
|
|
3925 |
apply(induct r arbitrary: c v rule: rexp.induct)
|
|
|
3926 |
apply(simp)
|
|
|
3927 |
apply(erule Prf.cases)
|
|
|
3928 |
apply(simp_all)[5]
|
|
|
3929 |
apply(simp)
|
|
|
3930 |
apply(erule Prf.cases)
|
|
|
3931 |
apply(simp_all)[5]
|
|
|
3932 |
apply(simp)
|
|
|
3933 |
apply(case_tac "c = char")
|
|
|
3934 |
apply(simp)
|
|
|
3935 |
apply(erule Prf.cases)
|
|
|
3936 |
apply(simp_all)[5]
|
|
|
3937 |
apply(simp)
|
|
|
3938 |
apply(erule Prf.cases)
|
|
|
3939 |
apply(simp_all)[5]
|
|
|
3940 |
defer
|
|
|
3941 |
apply(simp)
|
|
|
3942 |
apply(erule Prf.cases)
|
|
|
3943 |
apply(simp_all)[5]
|
|
|
3944 |
apply(simp)
|
|
|
3945 |
apply(case_tac "nullable rexp1")
|
|
|
3946 |
apply(simp)
|
|
|
3947 |
apply(erule Prf.cases)
|
|
|
3948 |
apply(simp_all)[5]
|
|
|
3949 |
apply(auto)[1]
|
|
|
3950 |
apply(erule Prf.cases)
|
|
|
3951 |
apply(simp_all)[5]
|
|
|
3952 |
apply(auto)[1]
|
|
|
3953 |
apply (metis list.distinct(1) v4)
|
|
|
3954 |
apply(auto)[1]
|
|
|
3955 |
apply (metis mkeps_flat)
|
|
|
3956 |
apply(auto)
|
|
|
3957 |
apply(erule Prf.cases)
|
|
|
3958 |
apply(simp_all)[5]
|
|
|
3959 |
apply(auto)[1]
|
|
|
3960 |
apply(simp add: v4)
|
|
|
3961 |
done
|
|
|
3962 |
|
|
|
3963 |
text {*
|
|
|
3964 |
|
|
|
3965 |
HERE: Crucial lemma that does not go through in the sequence case.
|
|
|
3966 |
|
|
|
3967 |
*}
|
|
|
3968 |
lemma v5:
|
|
|
3969 |
assumes "\<turnstile> v : der c r" "POSIX v (der c r)"
|
|
|
3970 |
shows "POSIX (injval r c v) r"
|
|
|
3971 |
using assms
|
|
|
3972 |
apply(induct arbitrary: v rule: der.induct)
|
|
|
3973 |
(* NULL case *)
|
|
|
3974 |
apply(simp)
|
|
|
3975 |
apply(erule Prf.cases)
|
|
|
3976 |
apply(simp_all)[5]
|
|
|
3977 |
(* EMPTY case *)
|
|
|
3978 |
apply(simp)
|
|
|
3979 |
apply(erule Prf.cases)
|
|
|
3980 |
apply(simp_all)[5]
|
|
|
3981 |
(* CHAR case *)
|
|
|
3982 |
apply(simp)
|
|
|
3983 |
apply(case_tac "c = c'")
|
|
|
3984 |
apply(auto simp add: POSIX_def)[1]
|
|
|
3985 |
apply(erule Prf.cases)
|
|
|
3986 |
apply(simp_all)[5]
|
|
|
3987 |
oops
|
|
|
3988 |
*)
|
|
|
3989 |
|
|
|
3990 |
|
|
|
3991 |
end |